Geometry and proof · Ages 14–15

Rectangles and squares worksheet

Part of: Quadrilaterals

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Geometry and proof · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.ABCD is a rectangle whose diagonals cross at E, and AE = 5 cm. x =
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2.ABCD is a square. x =
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3.ABCD is a rectangle whose diagonals cross at E. x =
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4.ABCD is a rectangle whose diagonals cross at E. ∠EBA = °, ∠AEB = °, ∠EBC = °
Fill in each blank
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5.The rectangle ABCD's diagonals cross at E, with AE = (3y + 5) cm and BD = (7y + 5) cm. y =
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6.The rectangle ABCD's diagonals cross at E, and ∠AEB = 140°. ∠ACB =
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7.The rectangle ABCD's diagonals cross at E, and ∠AEB = 108°. ∠ACB =
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8.The rectangle ABCD's diagonals cross at E, with AE = (3y + 5) cm and BD = (7y + 2) cm. y =
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9.The rectangle ABCD's diagonals cross at E, with AE = (y + 3) cm and BD = (3y + 4) cm. y =
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10.The rectangle ABCD's diagonals cross at E, and ∠AEB = 126°. ∠ACB =
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11.The parallelogram ABCD has diagonals of 10 cm and 10 cm. ABCD is a rectangle.
True or false?
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12.The parallelogram ABCD has diagonals of 11 cm and 16 cm. ABCD is a rectangle.
True or false?
Answer:

13.A rectangular board ABCD with AB = 35 cm and BC = 120 cm is sawn along both diagonals, which cross at E. What is the perimeter of the triangular piece ABE?
Working:
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14.A rectangular square's diagonals are 46 m long, and a fountain stands where they cross. How far is it from each corner?
Working:
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15.A carpenter makes a door frame 240 mm by 100 mm. To check its corners are right angles he measures the diagonals: each should be mm, and where they cross should be mm from every corner.
Fill in each blank
Working:
Answer:
16.Two strings run across a rectangular frame along its diagonals, crossing at E with ∠AEB = 46°. ∠BEC = °, ∠ECB = °, ∠EBA = °.
Fill in each blank
Working:
Answer:
17.A carpenter makes a door frame 120 mm by 50 mm. To check its corners are right angles he measures the diagonals: each should be mm, and where they cross should be mm from every corner.
Fill in each blank
Working:
Answer:
18.A rectangular board ABCD with AB = 16 cm and BC = 30 cm is sawn along both diagonals, which cross at E. What is the perimeter of the triangular piece ABE?
Working:
Answer:

Show the answers and working

Rectangles and squares worksheet: Answers and working

1. 5 cm
A rectangle's diagonals are equal and bisect each other, so AE = BE = CE = DE: x = 5 (prop. of rectangle).
2. 45°
A square's diagonal bisects its right angles: x = 90° ÷ 2 = 45° (prop. of square).
3. 64°
EA = EB (prop. of rectangle), so ∠EBA = 58° (base ∠s, isos. △); x = 180° − 2 × 58° = 64°.
4. 57, 66, 33
EA = EB, so ∠EBA = 57°; ∠AEB = 180° − 2 × 57° = 66°; ∠EBC = 90° − 57° = 33°.
5. 5
BD = AC = 2 × AE: 2(3y + 5) = 7y + 5, so y = 5.
6. 70°
∠BEC = 180° − 140° = 40° (adj. ∠s on st. line); EB = EC (prop. of rectangle), so ∠ECB = (180° − 40°) ÷ 2 = 70°.
7. 54°
∠BEC = 180° − 108° = 72° (adj. ∠s on st. line); EB = EC (prop. of rectangle), so ∠ECB = (180° − 72°) ÷ 2 = 54°.
8. 8
BD = AC = 2 × AE: 2(3y + 5) = 7y + 2, so y = 8.
9. 2
BD = AC = 2 × AE: 2(y + 3) = 3y + 4, so y = 2.
10. 63°
∠BEC = 180° − 126° = 54° (adj. ∠s on st. line); EB = EC (prop. of rectangle), so ∠ECB = (180° − 54°) ÷ 2 = 63°.

11. True
A parallelogram is a rectangle exactly when its diagonals are equal; 10 cm and 10 cm must match.
12. False
A parallelogram is a rectangle exactly when its diagonals are equal; 11 cm and 16 cm must match.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 160 cm
AC = 35² + 120² = 125 cm; a rectangle's diagonals are equal and bisect each other, so AE = BE = 125 ÷ 2; perimeter = 35 + 125 = 160 cm.
14. 23 m
A rectangle's diagonals are equal and bisect each other: the crossing is 46 ÷ 2 = 23 m from every corner.
15. 260, 130
Diagonal = 100² + 240² = 260 mm (Pyth. thm.); a rectangle's diagonals are equal and bisect each other, so the crossing is 260 ÷ 2 mm from each corner.
16. 134, 23, 67
∠BEC = 180° − 46° = 134°; EB = EC, so ∠ECB = (180° − 134°) ÷ 2 = 23°; ∠EBA = 90° − ∠EBC = 67°.
17. 130, 65
Diagonal = 50² + 120² = 130 mm (Pyth. thm.); a rectangle's diagonals are equal and bisect each other, so the crossing is 130 ÷ 2 mm from each corner.
18. 50 cm
AC = 16² + 30² = 34 cm; a rectangle's diagonals are equal and bisect each other, so AE = BE = 34 ÷ 2; perimeter = 16 + 34 = 50 cm.

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