Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the distance from I to B C is 2 , what is the square of the length of B C ?

Correct answer: 80 aligned & I M=r=2 & I A=(r)/( (A)/(2)) & (A)/(2)=(r)/(I A)=(2)/(3) & (A)/(2)= 2 5 =(B M)/(A M) & 2 5 =(a)/(5) & B C=a=4 5 & B C^2=80 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let A B C be an isosceles triangle with A B=A C and incentre I . If...

Let

A B C

be an isosceles triangle with

A B=A C

and incentre

I

. If

A I=3

and the distance from

I

B C

is 2 , what is the square of the length of

B C

Let A B C be an isosceles triangle with A B=A C and incentre I . If A I=3 and the distance from I to B C is 2 , what is

Correct answer

Step-by-step solution

\begin{aligned} & I M=r=2 \\ & I A=\frac{r}{\sin \frac{A}{2}} \\ & \sin \frac{A}{2}=\frac{r}{I A}=\frac{2}{3} \\ & \tan \frac{A}{2}=\frac{2}{\sqrt{5}}=\frac{B M}{A M} \\ & \frac{2}{\sqrt{5}}=\frac{a}{5} \\ & B C=a=4 \sqrt{5} \\ & B C^2=80\end{aligned}

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