Question

The triangle $\triangle ABC$ is an isosceles triangle where $AB = 4\sqrt{2}$ and $\angle B$ is a right angle. If $I$ is the incenter of $\triangle ABC, then what is $BI$? Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1.$ a) $2 + 2\sqrt{2}$ b) $2\sqrt{2}$ c) $4 + 2\sqrt{2}$ d) $4\sqrt{2}$

Answer 1

The length of BI is 2√2.

Option (b) is true.

Explanation:

To find the length of BI, we can use the properties of an isosceles right triangle and the incenter.

Since triangle ABC is an isosceles triangle with angle B as a right angle, we know that angle A = angle C = 45 degrees.

The incenter I is located at the midpoint of the hypotenuse AB.

Therefore, BI is equal to half the length of AB.

Given that AB = 4√2, we have:

BI = 1/2 * AB

= 1/2 * 4√2

= 2√2.

Hence,

The length of BI is 2√2.

Option (b) is true.