Year 4: Missing corners and missing numbers
Part of the Year 4 coordinates and translation chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Missing corners and missing numbers
Sometimes you are given three corners of a shape and you must find the fourth. You can use what you know about the shape. Here are four clues.
Rectangle and square. The sides go straight across and straight up. So the missing corner lines up with one corner (same x-coordinate) and with another corner (same y-coordinate).
Parallelogram. Opposite sides are parallel and the same length. Find the move along one side, then use the same move for the opposite side.
Kite and isosceles triangle. These shapes are symmetrical. Find the mirror line (the line of symmetry). The missing corner is the mirror image of its partner: the same distance from the line on the other side, at the same height.
Missing numbers on a line. If two points are on a horizontal line, they have the same y-coordinate. If they are on a vertical line, they have the same x-coordinate. So (2, 5) and (7, ?) on a horizontal line gives ? = 5.
Key ideas
- Rectangle or square: the missing corner has the same x as one corner and the same y as another corner.
- Parallelogram: opposite sides make the same move. Use the move from one side on the opposite side.
- Kite or isosceles triangle: find the mirror line. The missing corner is the same distance from it on the other side, at the same height.
- Horizontal line: the y-coordinate stays the same. Vertical line: the x-coordinate stays the same.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A rectangle has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (2, 1)
B = (6, 1)
C = (6, 4) - Find the side joining D to A. In a rectangle the sides are straight up and down or flat. AD goes straight up or down, so D has the same x-coordinate as A.A = (2, 1), so the x-coordinate of D is 2.
- Find the side joining D to C. CD is a flat side, so D has the same y-coordinate as C.C = (6, 4), so the y-coordinate of D is 4.
- Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.D = (2, 4)
Check. The completed shape is a rectangle. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A square has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (1, 1)
C = (4, 4)
D = (1, 4) - Find the side joining B to C. In a square the sides are straight up and down or flat. CB goes straight up or down, so B has the same x-coordinate as C.C = (4, 4), so the x-coordinate of B is 4.
- Find the side joining B to A. AB is a flat side, so B has the same y-coordinate as A.A = (1, 1), so the y-coordinate of B is 1.
- Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.B = (4, 1)
- Check the sides. A square has four equal sides.The sides are 3, 3, 3, 3 long. They are all equal.
Check. The completed shape is a square. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. In a parallelogram the opposite sides are parallel and the same length. So the missing corner is the same move away from its neighbour as the opposite sides show.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (1, 2)
B = (5, 2)
C = (7, 5) - Find the move from B to C. In a parallelogram, opposite sides are parallel and the same length. BC and AD are opposite sides, so they are the same move.B (5, 2) to C (7, 5): 2 right, 3 up.
- Do the same move from A. D is the same move away from A as C is from B.A = (1, 2)
2 right, 3 up
D = (1 + 2, 2 + 3) = (3, 5) - Draw the sides and check. Join the new corner to the others.
Check. The completed shape is a parallelogram. ✓
Watch out. Move the same way both times: left or right first, then up or down.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A kite is symmetrical about a line through two of its corners. The missing corner is the mirror image of its partner corner.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (3, 1)
C = (3, 7)
D = (1, 3) - Find the line of symmetry. A kite is symmetrical. The corners A and C are in a straight up-and-down line, so that line is the mirror line.A = (3, 1) and C = (3, 7)
The mirror line is x = 3. - Reflect D in the mirror line. D is 2 to the left of the line. B is the same distance on the other side, at the same height.D = (1, 3) is 2 left of x = 3.
B is 2 right: x = 3 + 2 = 5
Same height: y = 3 - Write the missing corner and draw the sides. Join it to the other corners.B = (5, 3)
Check. The completed shape is a kite. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. The apex of an isosceles triangle is straight above the middle of the base. Find the middle of the base, then use the height.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (2, 1)
B = (8, 1) - Find the middle of the base. The apex of an isosceles triangle is straight above (or below) the middle of the base AB.The base goes from x = 2 to x = 8.
The middle is (2 + 8) ÷ 2 = 5 - Use the height you are told. The question says C is 5 up.C = (5, 5)
- Draw the sides and check. Join the new corner to A and B. The two sloping sides match.
Check. The mirror line x = 5 cuts the base in half, so the two sloping sides are the same. ✓
Watch out. The middle of the base is halfway between the two x-coordinates, not one of them.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a horizontal line all have the same y-coordinate. So the missing number is the same as the y-coordinate you already know.
- Look at the line. P and Q are on a horizontal (flat) line.P = (2, 5)
- Use what points on a horizontal line have in common. Every point on this line has the same y-coordinate (the second number).The y-coordinate is 5.
- Fill in the missing number. The missing number is the second number of Q.Q = (7, 5)
Check. Both points have y = 5. ✓
Watch out. On a horizontal line the y-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a vertical line all have the same x-coordinate. So the missing number is the same as the x-coordinate you already know.
- Look at the line. P and Q are on a vertical (straight up) line.P = (4, 6)
- Use what points on a vertical line have in common. Every point on this line has the same x-coordinate (the first number).The x-coordinate is 4.
- Fill in the missing number. The missing number is the first number of Q.Q = (4, 2)
Check. Both points have x = 4. ✓
Watch out. On a vertical line the x-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A rectangle has straight sides. Use the corners you know to find the missing number in C.
- Write the corners. A rectangle ABCD has corners A, B, C, D in order.A (1, 2), B (6, 2), C (6, ?), D (1, 5)
- Look at the side DC. D and C are joined by a flat side, so they have the same y-coordinate.D is (1, 5), so the y-coordinate of C is 5.
- Check the side BC. B and C are joined by an upright side, so they have the same x-coordinate.B is (6, 2) and C is (6, ?). The x-coordinates match: 6.
Check. C = (6, 5). AB = 5 and DC = 5. AD = 3 and BC = 3. Opposite sides are equal. ✓
Watch out. Flat sides keep the y-coordinate. Upright sides keep the x-coordinate.
Watch out
Do not guess the corner by eye. Use the numbers. For a rectangle, copy one number from each of two corners. For a parallelogram, the missing corner does not line up with the other corners, so use the move (for example 2 right and 3 up) instead.Exercise 10C Finding missing corners and missing numbers
Non-calculatorKey words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A rectangle has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (1, 2)
B = (5, 2)
C = (5, 5) - Find the side joining D to A. In a rectangle the sides are straight up and down or flat. AD goes straight up or down, so D has the same x-coordinate as A.A = (1, 2), so the x-coordinate of D is 1.
- Find the side joining D to C. CD is a flat side, so D has the same y-coordinate as C.C = (5, 5), so the y-coordinate of D is 5.
- Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.D = (1, 5)
Check. The completed shape is a rectangle. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A rectangle has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.
- Write the coordinates of the corners you know. We use them to work out the missing corner.B = (7, 1)
C = (7, 4)
D = (2, 4) - Find the side joining A to D. In a rectangle the sides are straight up and down or flat. DA goes straight up or down, so A has the same x-coordinate as D.D = (2, 4), so the x-coordinate of A is 2.
- Find the side joining A to B. BA is a flat side, so A has the same y-coordinate as B.B = (7, 1), so the y-coordinate of A is 1.
- Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.A = (2, 1)
Check. The completed shape is a rectangle. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A square has sides that go straight across and straight up. The missing corner lines up with one corner going up or down, and with another corner going across.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (3, 2)
B = (6, 2)
D = (3, 5) - Find the side joining C to B. In a square the sides are straight up and down or flat. BC goes straight up or down, so C has the same x-coordinate as B.B = (6, 2), so the x-coordinate of C is 6.
- Find the side joining C to D. DC is a flat side, so C has the same y-coordinate as D.D = (3, 5), so the y-coordinate of C is 5.
- Write the missing corner and draw the sides. Put the x-coordinate first, then the y-coordinate. Join it to the other corners.C = (6, 5)
- Check the sides. A square has four equal sides.The sides are 3, 3, 3, 3 long. They are all equal.
Check. The completed shape is a square. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. In a parallelogram the opposite sides are parallel and the same length. So the missing corner is the same move away from its neighbour as the opposite sides show.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (2, 1)
B = (6, 1)
C = (8, 4) - Find the move from B to C. In a parallelogram, opposite sides are parallel and the same length. BC and AD are opposite sides, so they are the same move.B (6, 1) to C (8, 4): 2 right, 3 up.
- Do the same move from A. D is the same move away from A as C is from B.A = (2, 1)
2 right, 3 up
D = (2 + 2, 1 + 3) = (4, 4) - Draw the sides and check. Join the new corner to the others.
Check. The completed shape is a parallelogram. ✓
Watch out. Move the same way both times: left or right first, then up or down.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. In a parallelogram the opposite sides are parallel and the same length. So the missing corner is the same move away from its neighbour as the opposite sides show.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (1, 1)
C = (6, 5)
D = (2, 5) - Find the move from D to A. In a parallelogram, opposite sides are parallel and the same length. DA and CB are opposite sides, so they are the same move.D (2, 5) to A (1, 1): 1 left, 4 down.
- Do the same move from C. B is the same move away from C as A is from D.C = (6, 5)
1 left, 4 down
B = (6 − 1, 5 − 4) = (5, 1) - Draw the sides and check. Join the new corner to the others.
Check. The completed shape is a parallelogram. ✓
Watch out. Move the same way both times: left or right first, then up or down.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. A kite is symmetrical about a line through two of its corners. The missing corner is the mirror image of its partner corner.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (4, 1)
C = (4, 7)
D = (2, 3) - Find the line of symmetry. A kite is symmetrical. The corners A and C are in a straight up-and-down line, so that line is the mirror line.A = (4, 1) and C = (4, 7)
The mirror line is x = 4. - Reflect D in the mirror line. D is 2 to the left of the line. B is the same distance on the other side, at the same height.D = (2, 3) is 2 left of x = 4.
B is 2 right: x = 4 + 2 = 6
Same height: y = 3 - Write the missing corner and draw the sides. Join it to the other corners.B = (6, 3)
Check. The completed shape is a kite. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. The apex of an isosceles triangle is straight above the middle of the base. Find the middle of the base, then use the height.
- Write the coordinates of the corners you know. We use them to work out the missing corner.A = (1, 2)
B = (7, 2) - Find the middle of the base. The apex of an isosceles triangle is straight above (or below) the middle of the base AB.The base goes from x = 1 to x = 7.
The middle is (1 + 7) ÷ 2 = 4 - Use the height you are told. The question says C is 6 up.C = (4, 6)
- Draw the sides and check. Join the new corner to A and B. The two sloping sides match.
Check. The mirror line x = 4 cuts the base in half, so the two sloping sides are the same. ✓
Watch out. The middle of the base is halfway between the two x-coordinates, not one of them.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. An isosceles triangle is symmetrical about a line through its apex. The missing corner is the mirror image of its partner corner.
- Write the coordinates of the corners you know. We use them to work out the missing corner.B = (7, 2)
C = (4, 6) - Find the line of symmetry. The apex C is the top of the isosceles triangle. The line going straight down from it is the mirror line.C = (4, 6), so the mirror line is x = 4.
- Reflect B in the mirror line. B is 3 to the right of the line. A is the same distance on the other side, at the same height.B = (7, 2) is 3 right of x = 4.
A is 3 left: x = 4 − 3 = 1
Same height: y = 2 - Write the missing corner and draw the sides. Join it to the other corners.A = (1, 2)
Check. The completed shape is an isosceles triangle. ✓
Watch out. Check each side of your shape. It must look like the shape in the question.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a horizontal line all have the same y-coordinate. So the missing number is the same as the y-coordinate you already know.
- Look at the line. P and Q are on a horizontal (flat) line.P = (3, 4)
- Use what points on a horizontal line have in common. Every point on this line has the same y-coordinate (the second number).The y-coordinate is 4.
- Fill in the missing number. The missing number is the second number of Q.Q = (8, 4)
Check. Both points have y = 4. ✓
Watch out. On a horizontal line the y-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a horizontal line all have the same y-coordinate. So the missing number is the same as the y-coordinate you already know.
- Look at the line. P and Q are on a horizontal (flat) line.Q = (1, 7)
- Use what points on a horizontal line have in common. Every point on this line has the same y-coordinate (the second number).The y-coordinate is 7.
- Fill in the missing number. The missing number is the second number of P.P = (6, 7)
Check. Both points have y = 7. ✓
Watch out. On a horizontal line the y-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a vertical line all have the same x-coordinate. So the missing number is the same as the x-coordinate you already know.
- Look at the line. P and Q are on a vertical (straight up) line.P = (5, 2)
- Use what points on a vertical line have in common. Every point on this line has the same x-coordinate (the first number).The x-coordinate is 5.
- Fill in the missing number. The missing number is the first number of Q.Q = (5, 8)
Check. Both points have x = 5. ✓
Watch out. On a vertical line the x-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Points on a vertical line all have the same x-coordinate. So the missing number is the same as the x-coordinate you already know.
- Look at the line. P and Q are on a vertical (straight up) line.Q = (9, 1)
- Use what points on a vertical line have in common. Every point on this line has the same x-coordinate (the first number).The x-coordinate is 9.
- Fill in the missing number. The missing number is the first number of P.P = (9, 6)
Check. Both points have x = 9. ✓
Watch out. On a vertical line the x-coordinate stays the same. The other number changes.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Check Mei’s corner. A rectangle has a corner that is in line with two other corners, going straight across and straight up.
- Find what D must line up with. D is joined to A by an upright side, so D has the same x-coordinate as A. D is joined to C by a flat side, so D has the same y-coordinate as C.A = (1, 1), so x = 1.
C = (4, 5), so y = 5. - Write the right corner. Put x first, then y.D = (1, 5)
- Say what Mei did wrong. Mei wrote (1, 4). She used the y-coordinate 4 instead of 5.Mei’s point (1, 4) is not in line with C. The side DC would slope.
Check. AD = 4 and BC = 4, AB = 3 and DC = 3. Opposite sides are equal. ✓
Watch out. Copy the y-coordinate from C, not from somewhere else.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Think about examples. Is the sentence true for all of them, some of them, or none?
- Think about a rectangle. Its sides are straight across and straight up.Corners that are joined by an upright side have the same x-coordinate.
- Think about the missing corner. It is joined to two corners, one by a flat side and one by an upright side.It copies the x-coordinate of one and the y-coordinate of the other.
- Decide. This is always true.
Check. In the rectangle (2, 1), (6, 1), (6, 4), (2, 4), the corner (2, 4) uses x from (2, 1) and y from (6, 4). ✓
Watch out. Try some examples to test the sentence.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Think about examples. Is the sentence true for all of them, some of them, or none?
- Try a parallelogram that is a rectangle. A rectangle is a special parallelogram.Rectangle: the missing corner lines up with other corners.
- Try a slanted parallelogram. A parallelogram with slanted sides, like (1, 1), (5, 1), (7, 4), (3, 4).D = (3, 4) has x = 3, but A has x = 1 and B has x = 5. D is not straight above A.
- Decide. It happens for some parallelograms but not for all of them.This is sometimes true.
Check. For the rectangle it is true. For the slanted one D is not straight above A. ✓
Watch out. Try some examples to test the sentence.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. Think about examples. Is the sentence true for all of them, some of them, or none?
- Think about two points on a vertical line. They are straight above each other.They have the same x-coordinate.
- Test it. Try (4, 1) and (4, 7).Both have x = 4.
- Decide. This is always true.
Check. (9, 2) and (9, 8) are on a vertical line, and both have x = 9. ✓
Watch out. Try some examples to test the sentence.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other. The perimeter is the distance all the way round.
Method. First find the missing corner D. Then find the length and the width and work out the perimeter.
- Find D. D lines up with A (same x) and with C (same y).D = (2, 5)
- Find the length and the width. Count along AB and up BC.AB goes from x = 2 to x = 8: 6 cm.
BC goes from y = 2 to y = 5: 3 cm. - Add all four sides. The perimeter is the distance all the way round.6 + 3 + 6 + 3 = 18 cm
Check. 2 × (6 + 3) = 2 × 9 = 18. ✓
Watch out. Find D first. Then add all four sides, not only two.
Key words. A vertex is a corner of a shape. Horizontal means flat, like the horizon. Vertical means straight up and down. A parallelogram has two pairs of parallel sides. A kite has two pairs of equal sides that are next to each other.
Method. AB is one side of the square. Find how long it is. The other sides are the same length.
- Find the length of AB. A and B are on a vertical line, so count the difference in the y-coordinates.AB goes from y = 2 to y = 6: 6 − 2 = 4.
- Find where the other two corners are. The sides of a square are all 4 long. The other side from A goes across 4. We can go right (x = 3 + 4 = 7) or left (x = 3 − 4 = −1). The grid only has positive numbers, so go right.C and D have x = 7.
- Write the corners. C is level with B, and D is level with A.C = (7, 6) and D = (7, 2)
Check. All four sides are 4 long and the corners are right angles. ✓
Watch out. On this grid there are no negative numbers, so the square must go to the right.