Believe in yourself, Tommy!

Geometry Level 4

During his first week of Geometry class, Tommy has to draw an isosceles triangle, $\triangle ABC$ . He is only given two measurements:

$\angle ABC$ has a measure of $50°$ .
$B=50\text{ cm}$ .

Given the following information and assuming that a triangle is different if the angles are in a different order, how many different isosceles triangles could he draw?

2 1 4 3 Infinitely many

1 solution

A Former Brilliant Member
Jun 9, 2016

There are 3 cases to consider.

Case 1: A=C

Using the cosine rule, $50=\sqrt{2A^2-2A^2\cos{50^\circ}}\\2A^2(1-\cos{50^\circ})=2500\\A=C\approx59.155$

Checking, this does not violate the triangle inequality, so we have one possibility.

Case 2: A=B

This means that $\angle A=\angle B=50^\circ,\quad \angle C=40^\circ$ . Checking, we realise that this also results in one possibility.

Case 2: C=B

This means that $\angle C=\angle B=50^\circ,\quad \angle A=40^\circ$ . Checking, we realise that this also results in one possibility.

Thus, there are 3 possibilities in total.

Believe in yourself, Tommy!

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