+1253 The triangle \(\Delta ABC\) is an isosceles triangle where \(AB=4\sqrt2\) and \(\angle B\) is a right angle. If \(L\) is the incenter of \(\Delta ABC\), then what is \(BL\)?
+128475
We can use the Law of Sines to solve this
Since B is the right angle and the triangle is isosceles....then angles A and C = 45°
The incenter of the triangle is the point where the three angle bisectors meet
One of the triangles formed by their intersection is ΔBAL
Since B is bisected....then angle ABL = 45°
And since A is bisected....angle BAL = 22.5°
And angle ALB = 180 - 45 - 22.5 = 112.5°
So
BL AB
_______ = _______
sin BAL sin ALB
BL 4sqrt(2)
____ = _________
sin 22.5 sin 112.5
4sqrt (2) sin 22.5
BL = _______________ ≈ 2.34
sin 112.5
