GCSE Maths Transformations and Similarity (Higher): check-up and exam-style questions

Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.

Method. To reflect a shape, reflect every vertex over the mirror line x = 0, then join the image points in the same order.

1. Draw the mirror line. The line x = 0 is a vertical line: every point on it has x = 0. Everything in the picture will flip over this line.
   Draw a vertical line through x = 0.
2. Reflect each vertex of A in turn. For a reflection in a vertical line x = 0, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.
   (1, 2): 1 right of the line, so the image is 1 left of the line. x = 0 − 1 = −1. Image (−1, 2)
   (4, 2): 4 right of the line, so the image is 4 left of the line. x = 0 − 4 = −4. Image (−4, 2)
   (3, 4): 3 right of the line, so the image is 3 left of the line. x = 0 − 3 = −3. Image (−3, 4)
   (1, 4): 1 right of the line, so the image is 1 left of the line. x = 0 − 1 = −1. Image (−1, 4)
3. Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.
   B has vertices at (−1, 2), (−4, 2), (−3, 4), (−1, 4).

Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (1, 2) and (−1, 2) is (1 + −1) ÷ 2, (2 + 2) ÷ 2 = (0, 2), which lies on the line x = 0. ✓

Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = 0, so distances start from there.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (0, 0)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (2, 1): (2, 1) means 2 right, 1 up
   (4, 1): (4, 1) means 4 right, 1 up
   (2, 4): (2, 4) means 2 right, 4 up
3. Turn each move by 90° anticlockwise. A quarter turn anticlockwise changes a move using the rule (a, b) → (−b, a). Distances from C stay the same; only the direction changes.
   (2, 1) → (−1, 2)
   (4, 1) → (−1, 4)
   (2, 4) → (−4, 2)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (−1, 2) = (−1, 2)
   C + (−1, 4) = (−1, 4)
   C + (−4, 2) = (−4, 2)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (−1, 2), (−1, 4), (−4, 2).

Check. After a quarter turn the line from C to (2, 4) and the line from C to its image (−4, 2) are at right angles, so the "dot product" of the two moves is 0: (2 × −4) + (4 × 2) = 0. Both are also the same distance from C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).

Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.

1. Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.
   −3
   4 means 3 left, 4 up.
2. Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.
   (1, 1) → (1 − 3, 1 + 4) = (−2, 5)
   (3, 1) → (3 − 3, 1 + 4) = (0, 5)
   (3, 2) → (3 − 3, 2 + 4) = (0, 6)
   (2, 2) → (2 − 3, 2 + 4) = (−1, 6)
   (2, 4) → (2 − 3, 4 + 4) = (−1, 8)
   (1, 4) → (1 − 3, 4 + 4) = (−2, 8)
3. Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.
   B has vertices at (−2, 5), (0, 5), (0, 6), (−1, 6), (−1, 8), (−2, 8).

Check. Undo the translation: take −3
4 away from (−2, 5) and we get back to (−2 + 3, 5 − 4) = (1, 1). ✓

Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.

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.

1. Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.
   C = (1, 1)
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).
   (1, 1): (0, 0) = no move
   (3, 1): (2, 0) = 2 right
   (1, 4): (0, 3) = 3 up
3. 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)
   (2, 0) × 2 = (4, 0)
   (0, 3) × 2 = (0, 6)
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 + (4, 0) = (5, 1)
   C + (0, 6) = (1, 7)
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), (5, 1), (1, 7).

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. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.

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).

1. Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection.
2. Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.
   (1, 1) and (9, 1): midpoint = ((1 + 9) ÷ 2, (1 + 1) ÷ 2) = (5, 1)
   (4, 1) and (6, 1): midpoint = ((4 + 6) ÷ 2, (1 + 1) ÷ 2) = (5, 1)
3. Write the equation of the mirror line. The midpoints all have x = 5. The points are in a vertical line, so the mirror line is the vertical line x = 5.
   Mirror line: x = 5
4. Write the full description. We have found every piece of information needed for this transformation.
   Reflection in the line x = 5

Check. Reflect (4, 1) in x = 5 and we get (6, 1), which matches (6, 1) in B. ✓

Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

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).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.
   Distance² of (2, 2) from (0, 0) = 2² + 2² = 8
   Distance² of (2, −2) from (0, 0) = 2² + −2² = 8
   They match, and the same works for the other vertices, so the centre is (0, 0).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (2, 2): (2, 2) (2 right, 2 up)
   From C to (2, −2): (2, −2) (2 right, 2 down)
   This matches (a, b) → (b, −a), so the turn is 90° clockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° clockwise, centre (0, 0)

Check. Rotate (4, 2) about (0, 0) by 90° clockwise: we get (2, −4), which matches the vertex (2, −4) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Key words. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.

Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.

1. Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.
   Triangle 1: angle B = 30°, angle C = 80°, BC = 7 cm
   Triangle 2: angle Q = 30°, angle R = 80°, QR = 7 cm
2. Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.
   angle B = angle Q = 30°
   BC = QR = 7 cm (between the angles)
   angle C = angle R = 80°
3. Decide which condition is satisfied. ASA (angle, side, angle): two angles and the side between them are equal.
   The three equal parts make ASA.
4. State the conclusion with its reason. In an exam, name the condition as your reason.
   The triangles are congruent (ASA).

Check. The third angles are 180 − 30 − 80 = 70 in both triangles. ✓

Watch out. The side must be between the two angles.

Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.

Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.

1. Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.
   Known pair: 10 cm (small shape) and 15 cm (large shape).
   The side we want is the one marked x.
2. Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.
   15 cm ÷ 10 cm = 32
   So the large shape is 32 times as big as the small shape.
3. Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.
   x = 8 cm × 32 = 12 cm
4. Write the answer with its unit. Lengths need a unit.
   x = 12 cm

Check. Ratio check: 12 cm ÷ 8 cm = 32, the same scale factor as before. ✓

Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.

Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k² and the volume scale factor is k³. These work only for similar shapes or solids.

Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k².

1. Write down the length scale factor. It is the number that every length is multiplied by.
   k = 5
2. Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k².
   area scale factor = 5² = 5 × 5 = 25
3. Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.
   4 × 25 = 100 cm²

Check. Divide back: 100 ÷ 25 = 4. ✓

Watch out. Do not multiply the area or volume by k. Area uses k², volume uses k³.

Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k² and the volume scale factor is k³. These work only for similar shapes or solids.

Method. If the lengths of two similar shapes are in the ratio a : b, then the areas are in the ratio a² : b² and the volumes are in the ratio a³ : b³.

1. Write the length ratio in simplest form. Divide both numbers by their highest common factor.
   Lengths: 3 : 5
2. Square both numbers for the area ratio. The area scale factor is the square of the length scale factor.
   Areas: 3² : 5² = 9 : 25
3. Cube both numbers for the volume ratio. The volume scale factor is the cube of the length scale factor.
   Volumes: 3³ : 5³ = 27 : 125

Check. Test with a = 3, b = 5: a cube of side 3 has volume 27 and a cube of side 5 has volume 125. ✓

Watch out. Square (or cube) each number in the ratio, not just one of them.

Key words. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.

Method. To reflect a shape, reflect every vertex over the mirror line x = −1, then join the image points in the same order.

1. Draw the mirror line. The line x = −1 is a vertical line: every point on it has x = −1. Everything in the picture will flip over this line.
   Draw a vertical line through x = −1.
2. Reflect each vertex of A in turn. For a reflection in a vertical line x = −1, the y-coordinate stays the same. Count how far each point is from the line (sideways), then go the same distance on the other side.
   (2, 2): 3 right of the line, so the image is 3 left of the line. x = −1 − 3 = −4. Image (−4, 2)
   (5, 2): 6 right of the line, so the image is 6 left of the line. x = −1 − 6 = −7. Image (−7, 2)
   (2, 4): 3 right of the line, so the image is 3 left of the line. x = −1 − 3 = −4. Image (−4, 4)
3. Plot the image points and join them up in the same order to make shape B. The image has the same size and shape as the object but is the other way round, like a mirror image.
   B has vertices at (−4, 2), (−7, 2), (−4, 4).

Check. The mirror line should be exactly half-way between a point and its image. The midpoint of (2, 2) and (−4, 2) is (2 + −4) ÷ 2, (2 + 2) ÷ 2 = (−1, 2), which lies on the line x = −1. ✓

Watch out. Measure the distance from the mirror line, not from the y-axis. The line is x = −1, so distances start from there.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (0, 0)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (2, 2): (2, 2) means 2 right, 2 up
   (4, 2): (4, 2) means 4 right, 2 up
   (4, 3): (4, 3) means 4 right, 3 up
   (3, 3): (3, 3) means 3 right, 3 up
   (3, 5): (3, 5) means 3 right, 5 up
   (2, 5): (2, 5) means 2 right, 5 up
3. Turn each move by 90° anticlockwise. A quarter turn anticlockwise changes a move using the rule (a, b) → (−b, a). Distances from C stay the same; only the direction changes.
   (2, 2) → (−2, 2)
   (4, 2) → (−2, 4)
   (4, 3) → (−3, 4)
   (3, 3) → (−3, 3)
   (3, 5) → (−5, 3)
   (2, 5) → (−5, 2)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (−2, 2) = (−2, 2)
   C + (−2, 4) = (−2, 4)
   C + (−3, 4) = (−3, 4)
   C + (−3, 3) = (−3, 3)
   C + (−5, 3) = (−5, 3)
   C + (−5, 2) = (−5, 2)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (−2, 2), (−2, 4), (−3, 4), (−3, 3), (−5, 3), (−5, 2).

Check. After a quarter turn the line from C to (3, 5) and the line from C to its image (−5, 3) are at right angles, so the "dot product" of the two moves is 0: (3 × −5) + (5 × 3) = 0. Both are also the same distance from C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).

Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.

1. Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.
   −4
   3 means 4 left, 3 up.
2. Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.
   (1, 1) → (1 − 4, 1 + 3) = (−3, 4)
   (4, 2) → (4 − 4, 2 + 3) = (0, 5)
   (2, 4) → (2 − 4, 4 + 3) = (−2, 7)
3. Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.
   B has vertices at (−3, 4), (0, 5), (−2, 7).

Check. Undo the translation: take −4
3 away from (−3, 4) and we get back to (−3 + 4, 4 − 3) = (1, 1). ✓

Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.

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.

1. Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.
   C = (0, 0)
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).
   (2, 2): (2, 2) = 2 right, 2 up
   (4, 2): (4, 2) = 4 right, 2 up
   (2, 5): (2, 5) = 2 right, 5 up
3. Multiply each move by the scale factor 2. The image is bigger, so each distance from C is multiplied by 2.
   (2, 2) × 2 = (4, 4)
   (4, 2) × 2 = (8, 4)
   (2, 5) × 2 = (4, 10)
4. Add each new move to C to find the image vertex. The new moves are still measured from C.
   C + (4, 4) = (4, 4)
   C + (8, 4) = (8, 4)
   C + (4, 10) = (4, 10)
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 (4, 4), (8, 4), (4, 10).

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. A reflection flips a shape over a mirror line. The object and its image are the same distance from the mirror line, on opposite sides.

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).

1. Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, but B is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection.
2. Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.
   (2, 1) and (−4, 1): midpoint = ((2 + −4) ÷ 2, (1 + 1) ÷ 2) = (−1, 1)
   (4, 1) and (−6, 1): midpoint = ((4 + −6) ÷ 2, (1 + 1) ÷ 2) = (−1, 1)
3. Write the equation of the mirror line. The midpoints all have x = −1. The points are in a vertical line, so the mirror line is the vertical line x = −1.
   Mirror line: x = −1
4. Write the full description. We have found every piece of information needed for this transformation.
   Reflection in the line x = −1

Check. Reflect (4, 1) in x = −1 and we get (−6, 1), which matches (−6, 1) in B. ✓

Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

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).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 2 long in A and 2 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.
   Distance² of (2, 2) from (0, 0) = 2² + 2² = 8
   Distance² of (−2, 2) from (0, 0) = −2² + 2² = 8
   They match, and the same works for the other vertices, so the centre is (0, 0).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (2, 2): (2, 2) (2 right, 2 up)
   From C to (−2, 2): (−2, 2) (2 left, 2 up)
   This matches (a, b) → (−b, a), so the turn is 90° anticlockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° anticlockwise, centre (0, 0)

Check. Rotate (4, 2) about (0, 0) by 90° anticlockwise: we get (−2, 4), which matches the vertex (−2, 4) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

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).

1. 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 8 long in A and 4 long in B.
   The sides have changed length but the shape is the same, so the transformation is an enlargement.
2. Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.
   A side of A is 8 and the matching side of B is 4.
   4 ÷ 8 = 12
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 smaller than A, so the scale factor is a fraction between 0 and 1.
   Scale factor = 12
4. Find the centre of enlargement. A vertex that has not moved can only be the centre of enlargement.
   The vertex (9, 1) is in the same place on both shapes, so the centre is (9, 1).
5. Write the full description. We have found every piece of information needed for this transformation.
   Enlargement, scale factor 12, centre (9, 1)

Check. Enlarge (1, 1) from (9, 1) with scale factor 12: (9 + 12 × −8, 1 + 12 × 0) = (5, 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. Two shapes are congruent if they are exactly the same size and shape (they may be turned or flipped). The included angle is the angle between two given sides. The hypotenuse is the side opposite the right angle.

Method. Triangles are congruent if one fits exactly on the other. For triangles, we only need to check one of four conditions: SSS, SAS, ASA or RHS.

1. Write down what is known about each triangle. We can only use information that is given in the question or can be worked out from it.
   Triangle 1: AB = 6 cm, AC = 9 cm, angle A = 35°
   Triangle 2: PQ = 6 cm, PR = 9 cm, angle P = 35°
2. Match the equal parts. Parts in the same position (corresponding parts) must be equal in both triangles.
   AB = PQ = 6 cm
   angle A = angle P = 35° (between the sides)
   AC = PR = 9 cm
3. Decide which condition is satisfied. SAS (side, angle, side): two sides and the angle between them (the included angle) are equal.
   The three equal parts make SAS.
4. State the conclusion with its reason. In an exam, name the condition as your reason.
   The triangles are congruent (SAS).

Check. The angle 35° is between the 6 cm and the 9 cm sides in both triangles. ✓

Watch out. Say that the angle is between the equal sides.

Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.

Method. Similar shapes have corresponding sides in the same ratio. Find the scale factor from a pair of matching sides we know, then use it on the side we want.

1. Match the corresponding sides. In similar shapes the sides that are in the same position match. Their lengths are in the same ratio, and that ratio is the scale factor.
   Known pair: 5 cm (small shape) and 10 cm (large shape).
   The side we want is the one marked x.
2. Find the scale factor from the pair of sides we know. Scale factor = length in the large shape ÷ matching length in the small shape.
   10 cm ÷ 5 cm = 2
   So the large shape is 2 times as big as the small shape.
3. Use the scale factor to find x. x is on the large shape and we know the matching side on the small shape. Going from small to large we multiply by the scale factor.
   x = 3 cm × 2 = 6 cm
4. Write the answer with its unit. Lengths need a unit.
   x = 6 cm

Check. Ratio check: 6 cm ÷ 3 cm = 2, the same scale factor as before. ✓

Watch out. Make sure you pair the matching sides (same position in both shapes). Multiply when going from the small shape to the large one and divide when going from large to small.

Key words. Two shapes are similar if one is an enlargement of the other: same shape, different size. Corresponding sides are in the same position in both shapes. The scale factor is the number you multiply lengths by to go from one shape to the other.

Method. The flagpole and the ruler make similar triangles with the sun’s rays. Match shadow with shadow and height with height.

1. Find the scale factor from the ruler to the flagpole. Scale factor = flagpole shadow ÷ ruler shadow.
   12 ÷ 0.8 = 15
2. Find the height of the flagpole. Heights match each other. Multiply the ruler’s height by the scale factor.
   1 × 15 = 15 m

Check. 1 ÷ 0.8 = 1.25 and 15 ÷ 12 = 1.25. ✓

Watch out. Marks: 1 for a correct scale factor, 1 for using it, 1 for the answer with a unit.

Key words. The length scale factor k says how many times longer every length becomes. The area scale factor is k² and the volume scale factor is k³. These work only for similar shapes or solids.

Method. For similar shapes, if the lengths are multiplied by k, the areas are multiplied by k².

1. Write down the length scale factor. It is the number that every length is multiplied by.
   k = 3
2. Work out the area scale factor. Area has two lengths in it (length × width), so it is multiplied by k twice: k².
   area scale factor = 3² = 3 × 3 = 9
3. Multiply the original area by the area scale factor. The new value is the old value multiplied by the scale factor.
   8 × 9 = 72 cm²

Check. Divide back: 72 ÷ 9 = 8. ✓

Watch out. Do not multiply the area or volume by k. Area uses k², volume uses k³.

Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.

Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.

1. Do the first transformation: rotation of 90° clockwise about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (b, −a), then add the centre back on. Do this to every vertex of shape A.
   Shape B:
   (1, 1): move from centre (1, 1) → turned (1, −1) → (1, −1)
   (4, 1): move from centre (4, 1) → turned (1, −4) → (1, −4)
   (1, 3): move from centre (1, 3) → turned (3, −1) → (3, −1)
2. Then do the second transformation: reflection in the line x = 0. Reflecting in a vertical line x = 0: the y-coordinate stays the same and the new x is 2 × 0 − x (the same distance on the other side of the line). Do this to every vertex of shape B.
   Shape C:
   (1, −1) → (2 × 0 − 1, −1) = (−1, −1)
   (1, −4) → (2 × 0 − 1, −4) = (−1, −4)
   (3, −1) → (2 × 0 − 3, −1) = (−3, −1)
3. Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.
   B (1, −1), (1, −4), (3, −1) then C (−1, −1), (−1, −4), (−3, −1)
4. Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.
   Compare a side: it is 3 long in A and 3 long in C.
   The size is unchanged, but C is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection.
5. Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.
   The coordinates have been swapped and both signs changed: (1, 1) became (−1, −1).
6. Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.
   Midpoint of (1, 1) and (−1, −1) = (0, 0), which lies on y = −x.
7. Write the equation of the mirror line. A reflection that swaps x and y and changes both signs is a reflection in the line y = −x.
   Mirror line: y = −x
8. Write the full description. We have found every piece of information needed for this transformation.
   Reflection in the line y = −x

Check. Reflect (4, 1) in y = −x and we get (−1, −4), which matches (−1, −4) in C. ✓

Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.

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.

1. Mark the centre of enlargement C. Every distance in an enlargement is measured from C. C itself does not move.
   C = (2, 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
3. 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)
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)
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 (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.

Method. Show that the triangles have equal angles, giving a reason for each pair.

1. Angle at X. AC and BD are straight lines that cross at X.
   Angle AXB = angle CXD (vertically opposite angles are equal)
2. Angle at A and C. AB is parallel to CD and AC crosses both lines.
   Angle XAB = angle XCD (alternate angles are equal)
3. Conclude. Two pairs of angles are equal, so the third pair must be equal as well (angles in a triangle add up to 180°).
   The triangles are similar (equal angles).

Check. Alternate angles also give angle XBA = angle XDC, so all three pairs of angles are equal. ✓

Watch out. Marks: 1 for each correct pair of equal angles with a reason, 1 for the conclusion.