Geometry and proof · Ages 14–15

Rhombuses 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 rhombus whose diagonals cross at E. x =
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2.ABCD is a rhombus whose diagonals cross at E. x =
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3.A rhombus has diagonals of 48 cm and 36 cm. Its side =
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4.In the rhombus ABCD, ∠ABC = 112°. ∠BAC = °, ∠ABD = °
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5.ABCD is a rhombus whose diagonals cross at E. x =
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6.A rhombus has diagonals of 42 cm and 144 cm. Its side =
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7.In the rhombus ABCD, ∠ABC = 128°. ∠BAC = °, ∠ABD = °
Fill in each blank
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8.In the rhombus ABCD, ∠ABC = 130°. ∠BAC = °, ∠ABD = °
Fill in each blank
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9.The rhombus ABCD has sides of 10 cm and its diagonal AC is 12 cm. Its area =
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10.The rhombus ABCD has sides of 10 cm and its diagonal AC is 16 cm. Its area =
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11.The rhombus ABCD has sides of 75 cm and its diagonal AC is 42 cm. Its area =
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13.A rhombus floor tile has diagonals of 24 cm and 20 cm. What is its area?
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14.A rhombus window has sides of 20 cm and one diagonal of 24 cm. How long is the other diagonal?
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15.A rhombus kite's two struts (its diagonals) are 12 dm and 16 dm, crossing at right angles at their middles. Each side is dm, and the string round the frame is dm long.
Fill in each blank
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16.A rhombus window lattice ABCD has ∠ABC = 110°. ∠BAD = °, ∠ABD = °, ∠BAC = °.
Fill in each blank
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17.A rhombus floor tile has diagonals of 34 cm and 28 cm. What is its area?
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18.A rhombus window lattice ABCD has ∠ABC = 50°. ∠BAD = °, ∠ABD = °, ∠BAC = °.
Fill in each blank
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Show the answers and working

Rhombuses worksheet: Answers and working

1. 90°
A rhombus's diagonals are at right angles: x = 90° (prop. of rhombus).
2. 25°
A rhombus's diagonal bisects its angle: x = ∠BAE = 25° (prop. of rhombus).
3. 30 cm
The diagonals bisect each other at right angles, so each right triangle has legs 48 ÷ 2 and 36 ÷ 2: side = (p2)² + (q2)² = 30 cm (Pyth. thm.).
4. 34, 56
AB = BC, so ∠BAC = (180° − 112°) ÷ 2 = 34° (base ∠s, isos. △); BD bisects ∠ABC, so ∠ABD = 56° (prop. of rhombus).
5. 52°
∠AEB = 90° (prop. of rhombus); x = 180° − 90° − 38° = 52° (∠ sum of △).
6. 75 cm
The diagonals bisect each other at right angles, so each right triangle has legs 42 ÷ 2 and 144 ÷ 2: side = (p2)² + (q2)² = 75 cm (Pyth. thm.).
7. 26, 64
AB = BC, so ∠BAC = (180° − 128°) ÷ 2 = 26° (base ∠s, isos. △); BD bisects ∠ABC, so ∠ABD = 64° (prop. of rhombus).
8. 25, 65
AB = BC, so ∠BAC = (180° − 130°) ÷ 2 = 25° (base ∠s, isos. △); BD bisects ∠ABC, so ∠ABD = 65° (prop. of rhombus).
9. 96 cm²
AE = 12 ÷ 2 and BE = 10² − AE² (Pyth. thm.), so BD = 2 × BE; area = AC × BD ÷ 2 = 96 cm².
10. 96 cm²
AE = 16 ÷ 2 and BE = 10² − AE² (Pyth. thm.), so BD = 2 × BE; area = AC × BD ÷ 2 = 96 cm².
11. 3024 cm²
AE = 42 ÷ 2 and BE = 75² − AE² (Pyth. thm.), so BD = 2 × BE; area = AC × BD ÷ 2 = 3024 cm².

12. A. diagonals at right angles
A rhombus's diagonals are at right angles; a rectangle's are equal but needn't be perpendicular.

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

13. 240 cm²
The diagonals cut the rhombus into four congruent right triangles: area = 24 × 20 ÷ 2 = 240 cm².
14. 32 cm
Half the diagonal = 24 ÷ 2; the other half = 20² − (24 ÷ 2)² (Pyth. thm.); the other diagonal = twice that = 32 cm.
15. 10, 40
A rhombus's diagonals bisect each other at right angles: side = (12 ÷ 2)² + (16 ÷ 2)² = 10 dm; perimeter = 4 × 10 = 40 dm.
16. 70, 55, 35
∠BAD = 180° − 110° = 70° (int. ∠s); BD bisects ∠ABC: ∠ABD = 55°; the diagonals are at right angles: ∠BAC = 90° − 55° = 35°.
17. 476 cm²
The diagonals cut the rhombus into four congruent right triangles: area = 34 × 28 ÷ 2 = 476 cm².
18. 130, 25, 65
∠BAD = 180° − 50° = 130° (int. ∠s); BD bisects ∠ABC: ∠ABD = 25°; the diagonals are at right angles: ∠BAC = 90° − 25° = 65°.

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