The Shape-Shifting Triangle

Geometry Level 3

If the angle A of triangle ABC is doubled and the lengths of AB and AC are kept the same, the area of the triangle remains the same. Find angle A (in degrees).

This sum is from the 1994 AMTI Screening Test


The answer is 60.

Nanayaranaraknas Vahdam by Nanayaranaraknas Vahdam , 22, India

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2 solutions

Take the triangle ABC.

By sine law,

Area of triangle ABC = (1/2) × \times b × \times c × \times sin ⁡ \sin A = (1/2) × \times b × \times c × \times sin ⁡ \sin 2A = (1/2) × \times b × \times c × \times 2 × \times sin ⁡ \sin A × \times cos ⁡ \cos A

Therefore, cos ⁡ \cos A = 1/2

Therefore, angle A = 60

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Agnes Fung
May 19, 2014

Area of triangle = 1 2 × \frac{1}{2} \times a b sin ⁡ \sin c

c c in this case = ∠ A \angle A

1 2 × \frac{1}{2} \times a b sin ⁡ ∠ A = 1 2 × \sin \angle A = \frac{1}{2} \times a b sin ⁡ ∠ 2 A \sin \angle 2A

sin ⁡ A = sin ⁡ 2 A \sin A = \sin 2A

∠ A = 60 \angle A = \boxed{60}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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