GCSE Maths: Enlargement (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Enlargement
An enlargement changes the size of a shape but keeps its shape. Every length is multiplied by the same number, called the scale factor, and the angles do not change. The image is similar to the object (not congruent, unless the scale factor is 1). To describe an enlargement fully you give the scale factor and the centre of enlargement.
The centre is the fixed point that distances are measured from. Take any vertex, draw a straight line from the centre through the vertex, and extend it so that the distance from the centre is multiplied by the scale factor. For a scale factor of 2, the image vertex is twice as far from the centre as the object vertex, on the same line. On a grid it is easiest to count: if a vertex is 3 right and 1 up from the centre, then for a scale factor of 2 the image is 6 right and 2 up from the centre.
Fractional scale factors. A scale factor between 0 and 1, such as 12 or 13, makes the shape smaller. This is still called an enlargement. For a scale factor of 12, every distance from the centre is halved. If a vertex is 6 right and 4 up from the centre, its image is 3 right and 2 up from the centre. The image is closer to the centre than the object.
Describing an enlargement. To find the scale factor, divide a length on the image by the matching length on the object. To find the centre, draw a straight line through each vertex of the object and its image; the lines cross at the centre. Remember that a fractional scale factor still makes an enlargement.
Negative scale factors (Higher). With a negative scale factor the image is on the opposite side of the centre, and turned upside down. The size of the number gives the stretch, and the minus sign sends each point through the centre to the other side. For a scale factor of −2 a vertex that is 3 right and 1 up from the centre goes to the point 6 left and 2 down from the centre.
Key ideas
- An enlargement changes the size of a shape, not its shape. Angles stay the same.
- To enlarge, multiply the distance from the centre of enlargement to each vertex by the scale factor.
- Scale factor = length on the image ÷ matching length on the object.
- A scale factor between 0 and 1 makes the shape smaller (still an enlargement).
- Higher: a negative scale factor puts the image on the opposite side of the centre.
- The centre of enlargement is the only invariant point.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 2 about C, multiply the distance from C to every vertex by 2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (1, 1) = 1 right, 1 up
(4, 1): (4, 1) = 4 right, 1 up
(1, 3): (1, 3) = 1 right, 3 up - Multiply each move by the scale factor 2. The image is bigger, so each distance from C is multiplied by 2.(1, 1) × 2 = (2, 2)
(4, 1) × 2 = (8, 2)
(1, 3) × 2 = (2, 6) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (2, 2) = (2, 2)
C + (8, 2) = (8, 2)
C + (2, 6) = (2, 6) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (2, 2), (8, 2), (2, 6).
Check. Compare one side: it is 3 in A and 6 in B, and 3 × 2 = 6 = 6. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 2 about C, multiply the distance from C to every vertex by 2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (1, 1)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (0, 0) = no move
(4, 1): (3, 0) = 3 right
(3, 3): (2, 2) = 2 right, 2 up
(1, 3): (0, 2) = 2 up - Multiply each move by the scale factor 2. The image is bigger, so each distance from C is multiplied by 2.(0, 0) × 2 = (0, 0)
(3, 0) × 2 = (6, 0)
(2, 2) × 2 = (4, 4)
(0, 2) × 2 = (0, 4) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (0, 0) = (1, 1)
C + (6, 0) = (7, 1)
C + (4, 4) = (5, 5)
C + (0, 4) = (1, 5) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (1, 1), (7, 1), (5, 5), (1, 5).
Check. Compare one side: it is 3 in A and 6 in B, and 3 × 2 = 6 = 6. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 12 about C, multiply the distance from C to every vertex by 12, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(2, 2): (2, 2) = 2 right, 2 up
(8, 2): (8, 2) = 8 right, 2 up
(2, 6): (2, 6) = 2 right, 6 up - Multiply each move by the scale factor 12. The image is smaller, so each distance from C is multiplied by 12.(2, 2) × 12 = (1, 1)
(8, 2) × 12 = (4, 1)
(2, 6) × 12 = (1, 3) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (1, 1) = (1, 1)
C + (4, 1) = (4, 1)
C + (1, 3) = (1, 3) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (1, 1), (4, 1), (1, 3).
Check. Compare one side: it is 6 in A and 3 in B, and 6 × 12 = 3 = 3. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A scale factor between 0 and 1 makes the shape smaller; it is still called an enlargement.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of B is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (1, 1) and (2, 2): y = x
Line through (4, 1) and (8, 2): y = 0.25x
They cross at (0, 0), so C = (0, 0). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 2, centre (0, 0)
Check. Enlarge (1, 1) from (0, 0) with scale factor 2: (0 + 2 × 1, 0 + 2 × 1) = (2, 2), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 6 long in A and 3 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 6 and the matching side of B is 3.
3 ÷ 6 = 12 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is smaller than A, so the scale factor is a fraction between 0 and 1.Scale factor = 12
- Find the centre of enlargement. A vertex that has not moved can only be the centre of enlargement.The vertex (7, 1) is in the same place on both shapes, so the centre is (7, 1).
- Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 12, centre (7, 1)
Check. Enlarge (1, 1) from (7, 1) with scale factor 12: (7 + 12 × −6, 1 + 12 × 0) = (4, 1), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor −2 about C, multiply the distance from C to every vertex by −2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (2, 2)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(3, 3): (1, 1) = 1 right, 1 up
(5, 3): (3, 1) = 3 right, 1 up
(3, 4): (1, 2) = 1 right, 2 up - Multiply each move by the scale factor −2. The scale factor is negative, so the image is on the opposite side of C (and turned upside down). Multiply each move by −2; the signs of both numbers flip as well as the size changing.(1, 1) × −2 = (−2, −2)
(3, 1) × −2 = (−6, −2)
(1, 2) × −2 = (−2, −4) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (−2, −2) = (0, 0)
C + (−6, −2) = (−4, 0)
C + (−2, −4) = (0, −2) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (0, 0), (−4, 0), (0, −2).
Check. Compare one side: it is 2 in A and 4 in B, and 2 × 2 = 4 = 4. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A negative scale factor puts the image on the other side of the centre.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of B is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. B is on the opposite side of the centre from A and is upside down, so the scale factor is negative.Scale factor = −2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (1, 1) and (1, 4): x = 1
Line through (4, 1) and (−5, 4): y = −13x + 73
They cross at (1, 2), so C = (1, 2). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor −2, centre (1, 2)
Check. Enlarge (1, 1) from (1, 2) with scale factor −2: (1 + −2 × 0, 2 + −2 × −1) = (1, 4), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Watch out
Measure from the centre of enlargement, not from the origin (unless that is the centre) and not from the shape’s own corner. Do not add the scale factor to the coordinates: you must multiply the distances from the centre. And a scale factor like 12 makes the shape smaller, but it is still an enlargement.Exercise 21D Enlargement
Non-calculatorKey words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 2 about C, multiply the distance from C to every vertex by 2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (1, 1) = 1 right, 1 up
(2, 1): (2, 1) = 2 right, 1 up
(1, 3): (1, 3) = 1 right, 3 up - Multiply each move by the scale factor 2. The image is bigger, so each distance from C is multiplied by 2.(1, 1) × 2 = (2, 2)
(2, 1) × 2 = (4, 2)
(1, 3) × 2 = (2, 6) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (2, 2) = (2, 2)
C + (4, 2) = (4, 2)
C + (2, 6) = (2, 6) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (2, 2), (4, 2), (2, 6).
Check. Compare one side: it is 2 in A and 4 in B, and 2 × 2 = 4 = 4. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 3 about C, multiply the distance from C to every vertex by 3, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (1, 1) = 1 right, 1 up
(2, 1): (2, 1) = 2 right, 1 up
(1, 3): (1, 3) = 1 right, 3 up - Multiply each move by the scale factor 3. The image is bigger, so each distance from C is multiplied by 3.(1, 1) × 3 = (3, 3)
(2, 1) × 3 = (6, 3)
(1, 3) × 3 = (3, 9) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (3, 3) = (3, 3)
C + (6, 3) = (6, 3)
C + (3, 9) = (3, 9) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (3, 3), (6, 3), (3, 9).
Check. Compare one side: it is 2 in A and 6 in B, and 2 × 3 = 6 = 6. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 2 about C, multiply the distance from C to every vertex by 2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (1, 1)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (0, 0) = no move
(2, 1): (1, 0) = 1 right
(1, 3): (0, 2) = 2 up - Multiply each move by the scale factor 2. The image is bigger, so each distance from C is multiplied by 2.(0, 0) × 2 = (0, 0)
(1, 0) × 2 = (2, 0)
(0, 2) × 2 = (0, 4) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (0, 0) = (1, 1)
C + (2, 0) = (3, 1)
C + (0, 4) = (1, 5) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (1, 1), (3, 1), (1, 5).
Check. Compare one side: it is 2 in A and 4 in B, and 2 × 2 = 4 = 4. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 3 about C, multiply the distance from C to every vertex by 3, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 1)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(1, 1): (1, 0) = 1 right
(2, 1): (2, 0) = 2 right
(1, 3): (1, 2) = 1 right, 2 up - Multiply each move by the scale factor 3. The image is bigger, so each distance from C is multiplied by 3.(1, 0) × 3 = (3, 0)
(2, 0) × 3 = (6, 0)
(1, 2) × 3 = (3, 6) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (3, 0) = (3, 1)
C + (6, 0) = (6, 1)
C + (3, 6) = (3, 7) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (3, 1), (6, 1), (3, 7).
Check. Compare one side: it is 2 in A and 6 in B, and 2 × 3 = 6 = 6. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 3 about C, multiply the distance from C to every vertex by 3, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (1, 1)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(2, 1): (1, 0) = 1 right
(3, 1): (2, 0) = 2 right
(3, 2): (2, 1) = 2 right, 1 up
(2, 2): (1, 1) = 1 right, 1 up - Multiply each move by the scale factor 3. The image is bigger, so each distance from C is multiplied by 3.(1, 0) × 3 = (3, 0)
(2, 0) × 3 = (6, 0)
(2, 1) × 3 = (6, 3)
(1, 1) × 3 = (3, 3) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (3, 0) = (4, 1)
C + (6, 0) = (7, 1)
C + (6, 3) = (7, 4)
C + (3, 3) = (4, 4) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (4, 1), (7, 1), (7, 4), (4, 4).
Check. Compare one side: it is 1 in A and 3 in B, and 1 × 3 = 3 = 3. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. Multiply the distance from C, not the coordinates themselves (unless C is the origin).
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 12 about C, multiply the distance from C to every vertex by 12, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(2, 2): (2, 2) = 2 right, 2 up
(8, 2): (8, 2) = 8 right, 2 up
(2, 6): (2, 6) = 2 right, 6 up - Multiply each move by the scale factor 12. The image is smaller, so each distance from C is multiplied by 12.(2, 2) × 12 = (1, 1)
(8, 2) × 12 = (4, 1)
(2, 6) × 12 = (1, 3) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (1, 1) = (1, 1)
C + (4, 1) = (4, 1)
C + (1, 3) = (1, 3) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (1, 1), (4, 1), (1, 3).
Check. Compare one side: it is 6 in A and 3 in B, and 6 × 12 = 3 = 3. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A scale factor between 0 and 1 makes the shape smaller; it is still called an enlargement.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 12 about C, multiply the distance from C to every vertex by 12, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(4, 2): (4, 2) = 4 right, 2 up
(8, 2): (8, 2) = 8 right, 2 up
(8, 6): (8, 6) = 8 right, 6 up
(4, 4): (4, 4) = 4 right, 4 up - Multiply each move by the scale factor 12. The image is smaller, so each distance from C is multiplied by 12.(4, 2) × 12 = (2, 1)
(8, 2) × 12 = (4, 1)
(8, 6) × 12 = (4, 3)
(4, 4) × 12 = (2, 2) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (2, 1) = (2, 1)
C + (4, 1) = (4, 1)
C + (4, 3) = (4, 3)
C + (2, 2) = (2, 2) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (2, 1), (4, 1), (4, 3), (2, 2).
Check. Compare one side: it is 4 in A and 2 in B, and 4 × 12 = 2 = 2. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A scale factor between 0 and 1 makes the shape smaller; it is still called an enlargement.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor 13 about C, multiply the distance from C to every vertex by 13, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (0, 0)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(3, 3): (3, 3) = 3 right, 3 up
(9, 3): (9, 3) = 9 right, 3 up
(3, 6): (3, 6) = 3 right, 6 up - Multiply each move by the scale factor 13. The image is smaller, so each distance from C is multiplied by 13.(3, 3) × 13 = (1, 1)
(9, 3) × 13 = (3, 1)
(3, 6) × 13 = (1, 2) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (1, 1) = (1, 1)
C + (3, 1) = (3, 1)
C + (1, 2) = (1, 2) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (1, 1), (3, 1), (1, 2).
Check. Compare one side: it is 6 in A and 2 in B, and 6 × 13 = 2 = 2. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A scale factor between 0 and 1 makes the shape smaller; it is still called an enlargement.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Enlarge about (0, 0): multiply each move from the centre by 3, then add the centre back on.Enlargement with scale factor 3 and centre (0, 0)
- Apply it to the point. Follow the rule using the coordinates of the point.(4, 2): move from centre (4, 2) × 3 = (12, 6) → (12, 6)
- Write the coordinates of the image. The image of P is called P'.P' = (12, 6)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Enlarge about (0, 0): multiply each move from the centre by 12, then add the centre back on.Enlargement with scale factor 12 and centre (0, 0)
- Apply it to the point. Follow the rule using the coordinates of the point.(4, 2): move from centre (4, 2) × 0.5 = (2, 1) → (2, 1)
- Write the coordinates of the image. The image of P is called P'.P' = (2, 1)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Enlarge about (1, −1): multiply each move from the centre by 2, then add the centre back on.Enlargement with scale factor 2 and centre (1, −1)
- Apply it to the point. Follow the rule using the coordinates of the point.(4, 2): move from centre (3, 3) × 2 = (6, 6) → (7, 5)
- Write the coordinates of the image. The image of P is called P'.P' = (7, 5)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Enlarge about (0, 0): multiply each move from the centre by 14, then add the centre back on.Enlargement with scale factor 14 and centre (0, 0)
- Apply it to the point. Follow the rule using the coordinates of the point.(4, 8): move from centre (4, 8) × 0.25 = (1, 2) → (1, 2)
- Write the coordinates of the image. The image of P is called P'.P' = (1, 2)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
Method. Every length is multiplied by the scale factor, so multiply both measurements by 2.5.
- Multiply the length. The 15 cm side becomes 2.5 times as long.15 × 2.5 = 37.5 cm
- Multiply the width. The 10 cm side becomes 2.5 times as long.10 × 2.5 = 25 cm
- Write the answer. Give the dimensions in the same order as the question.25 cm by 37.5 cm
Check. 37.5 ÷ 25 = 1.5 and 15 ÷ 10 = 1.5, so the shapes have the same proportions. ✓
Watch out. Multiply both lengths by the scale factor, not just one of them.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of B is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (1, 1) and (2, 2): y = x
Line through (4, 1) and (8, 2): y = 0.25x
They cross at (0, 0), so C = (0, 0). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 2, centre (0, 0)
Check. Enlarge (1, 1) from (0, 0) with scale factor 2: (0 + 2 × 1, 0 + 2 × 1) = (2, 2), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 6 long in A and 3 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 6 and the matching side of B is 3.
3 ÷ 6 = 12 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is smaller than A, so the scale factor is a fraction between 0 and 1.Scale factor = 12
- Find the centre of enlargement. A vertex that has not moved can only be the centre of enlargement.The vertex (7, 1) is in the same place on both shapes, so the centre is (7, 1).
- Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 12, centre (7, 1)
Check. Enlarge (1, 1) from (7, 1) with scale factor 12: (7 + 12 × −6, 1 + 12 × 0) = (4, 1), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 1 long in A and 3 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 1 and the matching side of B is 3.
3 ÷ 1 = 3 - Decide whether it is positive or negative. B is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 3
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (2, 1) and (4, 1): y = 1
Line through (2, 2) and (4, 4): y = x
They cross at (1, 1), so C = (1, 1). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 3, centre (1, 1)
Check. Enlarge (2, 1) from (1, 1) with scale factor 3: (1 + 3 × 1, 1 + 3 × 0) = (4, 1), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Method. An enlargement multiplies the distance from the centre by the scale factor. Look at what Sam did to each vertex.
- Look at Sam’s working. Sam went from (1, 1) to (4, 4) by adding 3 to each coordinate. That is a translation, not an enlargement.(1, 1) → (4, 4) is + 3, not × 3
- Do it properly. The centre is the origin, so the move from the centre is the point itself. Multiply both coordinates by 3.(1, 1) → (1 × 3, 1 × 3) = (3, 3)
(2, 1) → (2 × 3, 1 × 3) = (6, 3)
(1, 3) → (1 × 3, 3 × 3) = (3, 9) - Write the image. (3, 3), (6, 3), (3, 9)
Check. Sam’s base was 1 long and is now still 1 long, so his shape has not grown. The correct base is 3 long. ✓
Watch out. Enlargement multiplies; it does not add.
Method. Think about what the word “enlargement” means in maths: any transformation that multiplies lengths by a scale factor about a centre.
- Check the definition. An enlargement multiplies every distance from the centre by the scale factor. The scale factor can be any number, not only numbers bigger than 1.Scale factor 12: every length is halved.
- Compare. The image has the same shape as the object, so the transformation is an enlargement with a scale factor that makes things smaller.Tilly is wrong.
Check. An enlargement with scale factor 2 followed by one with scale factor 12 about the same centre gets us back to the start. ✓
Watch out. In examination questions you must still write “enlargement” for a fractional scale factor.
Method. Work out the sides of B, then use Pythagoras’ theorem to test for a right angle.
- Multiply each side by 2. Every length in an enlargement is multiplied by the scale factor.3 × 2 = 6, 4 × 2 = 8, 5 × 2 = 10
- Square the two shorter sides and add. If the triangle is right-angled, the sum of these squares equals the square of the longest side.62 + 82 = 36 + 64 = 100
- Compare with the longest side. 102 = 100, which is equal, so B has a right angle.
Check. An enlargement does not change angles, so the right angle of A is also in B. ✓
Watch out. Enlargements keep angles the same. Only the lengths change.
Method. Follow one point through the two enlargements.
- Enlarge (5, 4) with scale factor 2, centre (3, 1). Move from the centre: (5 − 3, 4 − 1) = (2, 3). Multiply by 2.(2, 3) × 2 = (4, 6), so the image is (3 + 4, 1 + 6) = (7, 7)
- Enlarge (7, 7) with scale factor 12, centre (3, 1). Move from the centre: (7 − 3, 7 − 1) = (4, 6). Multiply by 12.(4, 6) × 12 = (2, 3), so the image is (3 + 2, 1 + 3) = (5, 4)
- Conclude. The point is back at (5, 4). The same happens to every point, so C is the same as A.2 × 12 = 1: the two scale factors cancel.
Check. Multiplying the scale factors gives 2 × 12 = 1, and an enlargement with scale factor 1 does not change a shape. ✓
Watch out. Two enlargements about the same centre multiply their scale factors.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor −2 about C, multiply the distance from C to every vertex by −2, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (2, 2)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(3, 3): (1, 1) = 1 right, 1 up
(5, 3): (3, 1) = 3 right, 1 up
(3, 4): (1, 2) = 1 right, 2 up - Multiply each move by the scale factor −2. The scale factor is negative, so the image is on the opposite side of C (and turned upside down). Multiply each move by −2; the signs of both numbers flip as well as the size changing.(1, 1) × −2 = (−2, −2)
(3, 1) × −2 = (−6, −2)
(1, 2) × −2 = (−2, −4) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (−2, −2) = (0, 0)
C + (−6, −2) = (−4, 0)
C + (−2, −4) = (0, −2) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (0, 0), (−4, 0), (0, −2).
Check. Compare one side: it is 2 in A and 4 in B, and 2 × 2 = 4 = 4. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A negative scale factor puts the image on the other side of the centre.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. To enlarge a shape with scale factor −12 about C, multiply the distance from C to every vertex by −12, keeping the same line through C.
- Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.C = (3, 2)
- Find the move from C to each vertex. We need to know how far each point is from C, across and up (or down).(5, 2): (2, 0) = 2 right
(7, 2): (4, 0) = 4 right
(7, 6): (4, 4) = 4 right, 4 up - Multiply each move by the scale factor −12. The scale factor is negative, so the image is on the opposite side of C (and turned upside down). Multiply each move by −12; the signs of both numbers flip as well as the size changing.(2, 0) × −12 = (−1, 0)
(4, 0) × −12 = (−2, 0)
(4, 4) × −12 = (−2, −2) - Add each new move to C to find the image vertex. The new moves are still measured from C.C + (−1, 0) = (2, 2)
C + (−2, 0) = (1, 2)
C + (−2, −2) = (1, 0) - Plot the image points and join them to make shape B. The dashed lines from C pass through each vertex and its image: this is a quick way to check the picture.B has vertices at (2, 2), (1, 2), (1, 0).
Check. Compare one side: it is 4 in A and 2 in B, and 4 × 12 = 2 = 2. ✓
Watch out. Measure from the centre of enlargement, not from the origin and not from the edge of the shape. A negative scale factor puts the image on the other side of the centre.
Key words. An enlargement changes the size of a shape but not its shape. The scale factor says how many times bigger each length becomes. The centre of enlargement is the fixed point that distances are measured from.
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in B.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of B is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. B is on the opposite side of the centre from A and is upside down, so the scale factor is negative.Scale factor = −2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (1, 1) and (1, 4): x = 1
Line through (4, 1) and (−5, 4): y = −13x + 73
They cross at (1, 2), so C = (1, 2). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor −2, centre (1, 2)
Check. Enlarge (1, 1) from (1, 2) with scale factor −2: (1 + −2 × 0, 2 + −2 × −1) = (1, 4), which is the matching vertex of B. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
Method. Work out the image of (4, 3) under each transformation and compare.
- Enlargement, scale factor −1, centre (1, 1). Move from the centre: (4 − 1, 3 − 1) = (3, 2). Multiply by −1.(3, 2) × (−1) = (−3, −2), so the image is (1 − 3, 1 − 2) = (−2, −1)
- Rotation 180° about (1, 1). The move (3, 2) becomes (−3, −2).Image = (1 − 3, 1 − 2) = (−2, −1)
- Compare. The images are the same.Both give (−2, −1).
Check. In both cases the point goes straight through the centre to the other side, the same distance away. ✓
Watch out. Multiplying by −1 changes both signs, just like a half turn does.