Question

The short sides of a parallelogram are both 12.0 cm. The acute angles of the parallelogram are 65°, and the short diagonal is 15.0 cm. Determine the length of the long sides of the parallelogram. Round your answer to the nearest tenth of a centimetre.

Answer 1

15.4 cm

Explanation:

You want the long side of a parallelogram with short side 12 cm, short diagonal 15 cm, and acute angle 65°.

Law of sines

The law of sines can be used to find long side 'b' from short side 'a' and short diagonal 'd'. But first, we need to know the angle B opposite the long side in the triangle with sides a, b, d.

Angle A

Angle B can be found using the angle sum theorem if we can find the measure of acute angle A opposite side 'a'. The law of sines helps here:

sin(A)/a = sin(65°)/d

A = arcsin(a/d·sin(65°)) = arcsin(12/15·sin(65°)) ≈ 46.473°.

B = 180° -65° -46.473° ≈ 68.527°

Long side

Finally, side 'b' is found from the relation ...

b/sin(B) = d/sin(65°)

b = 15·sin(68.527°)/sin(65°) ≈ 15.402

The length of the long side of the parallelogram is about 15.4 cm.

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