A rhombus has diagonals of 30 cm and 72 cm. Its side = □

39 cm. The diagonals bisect each other at right angles, so each right triangle has legs 30 ÷ 2 and 72 ÷ 2: side = √((p/2)^2 + (q/2)^2) = 39 cm (Pyth. thm.).

Which of these does every rhombus have, but not every rectangle? A. diagonals at right angles B. equal diagonals C. diagonals bisecting each other D. both pairs of opposite sides parallel Choose the right answer

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

Quadrilaterals · Ages 14–15

Rhombuses

Diagonals at right angles, bisecting the angles: four congruent right triangles.

How to do it

Method

Diagonals at right angles, bisecting the angles: four congruent right triangles

∠ABC = 70°: ∠ABE = 35°, ∠BAE = 90° − 35° = 55°

Another way

An isosceles triangle

∠BAC = (180° − 70°) ÷ 2 = 55°

In full

A rhombus is a parallelogram with four equal sides. Besides a parallelogram's properties, its diagonals are at right angles and bisect its angles (prop. of rhombus), so they cut it into 4 congruent right-angled triangles. Chasing angles: ∠ABC = 70°, and the diagonal BD bisects it, so ∠ABE = 35°; ∠AEB = 90°, so ∠BAE = 90° − 35° = 55°. Another way: an isosceles triangle. AB = BC, so △ABC is isosceles and ∠BAC = (180° − 70°) ÷ 2 = 55°, the same. Diagonals of 6 cm and 8 cm: each right triangle has legs of 3 cm and 4 cm, so a side = 3² + 4² = 5 cm; the area = 6 × 8 ÷ 2 = 24 cm².

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
A rhombus has diagonals of 30 cm and 72 cm. Its side =
Show the answer and working
Answer: 39 cm
The diagonals bisect each other at right angles, so each right triangle has legs 30 ÷ 2 and 72 ÷ 2: side = (p2)² + (q2)² = 39 cm (Pyth. thm.).
Example 2
ABCD is a rhombus whose diagonals cross at E. x =
Show the answer and working
Answer: 27°
∠AEB = 90° (prop. of rhombus); x = 180° − 90° − 63° = 27° (∠ sum of △).
Example 3
Which of these does every rhombus have, but not every rectangle?
- A. diagonals at right angles
- B. equal diagonals
- C. diagonals bisecting each other
- D. both pairs of opposite sides parallel
Choose the right answer
Show the answer and working
Answer: A. diagonals at right angles
A rhombus's diagonals are at right angles; a rectangle's are equal but needn't be perpendicular.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.ABCD is a rhombus whose diagonals cross at E. x =
Answer:
2.A rhombus has diagonals of 6 cm and 8 cm. Its side =
Answer:
3.In the rhombus ABCD, ∠ABC = 62°. ∠BAC = °, ∠ABD = °
Fill in each blank
Answer:
4.The rhombus ABCD has sides of 10 cm and its diagonal AC is 12 cm. Its area =
Answer:
5.Which of these does every rhombus have, but not every rectangle?
- A. equal diagonals
- B. diagonals at right angles
- C. diagonals bisecting each other
- D. both pairs of opposite sides parallel
Choose the right answer
Answer:

All the answers

1. 30°
2. 5 cm
3. 59, 31
4. 96 cm²
5. B. diagonals at right angles

Free worksheet: RhombusesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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