A geometry problem by Ahmed Moh AbuBakr

Geometry Level 2

$ABC$ is an isosceles triangle such that $AB = AC$ and $\frac {\angle BAC}{\angle MCB} = \frac a b$ for coprime positive integers $a,b$ . What is the value of $a+b$ ?

The answer is 3.

1 solution

Brian Charlesworth
Apr 23, 2015

Let $P$ be the midpoint of $BC.$ Then $\Delta APB$ , $\Delta APC$ and $\Delta CMB$ are all similar, and thus $y = 2x \Longrightarrow \dfrac{y}{x} = 2.$

With $a,b$ coprime, we then have that $a + b = 2 + 1 = \boxed{3}.$

WE can write (90-y)+x=180-(90+x). This gives y=2x and y/x=2/1.Thus a+b=3.

Siddharth Singh - 6 years, 1 month ago

A geometry problem by Ahmed Moh AbuBakr

The answer is 3.

1 solution

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