Question

PLEASE HELPPP

ABCD is a rhombus in which AC = 16cm and BD = 12cm.
i) What is the magnitude of AÔB?
ii) Find the length of AB​

Answer 1

(i) 90°

(ii) 10 cm

Explanation:

What is a Rhombus?

A rhombus is a special parallelogram that has all four sides equal. But unlike a square, the angles are not right angles

Properties of a rhombus

1. Opposite angles are equal.
2. The opposite sides are equal and parallel.
3. Diagonals bisect each other at 90°
4. The sum of any two adjacent or consecutive angles is 180°.
   

We are given AC = 16 cm, BD = 12 cm
Using property 3, we see that

AO = CO = AB/2 = 16/2 = 8 cm

BO = DO =12/2 = 6 cm

m∠AOB = 90° Answer (i)

(ii) Since m∠AOB = 90°, ΔAOB is a right triangle with AB as the hypotenuse and AO, BO as the legs of the right triangle

By the Pythagorean Theorem
AB² = AO² + BO²

AB² = 8² + 6²

AB² = 64 + 36

AB² = 100

AB = √100 = 10 cm Answer (ii)

Answer 2

• ∠AOB = 90°
• AB = 10 cm

Explanation:

You have a rhombus ABCD with diagonals AC = 16 cm and BD = 12 cm. You want the magnitude of central angle AOB, and the length of one side, AB.

Rhombus

The diagonals of a rhombus bisect each other and cross at right angle. They divide the figure into four (4) congruent right triangles.

The central angle AOB has magnitude 90°.

Sides

Each of these right triangles will have leg lengths of 6 cm and 8 cm, which you recognize as having the ratio 3 : 4. These are the leg lengths of a 3:4:5 right triangle. In this rhombus, those right triangles have lengths 6 cm : 8 cm : 10 cm. The side AB of the rhombus is 10 cm.

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Additional comment

If you don't remember the 3-4-5 right triangle, you can use the Pythagorean theorem to find the hypotenuse of the triangle with legs 6 cm and 8 cm:

c² = a² +b²

c² = 6² +8² = 36 +64 = 100

c = √100 = 10

The side AB is 10 cm long.