Question

In an isosceles triangle △ABC with base of AB=8, points K and L are marked on sides AC and BC accordingly, such that KL∥ AB. It turns out that △BLK and △AKB are also isosceles triangles. What can be the length of CL?

Answer 1

CL = (KL × CB)/8

Explanation:

First we would draw the diagram obtained from the given information. Then find the relationship that enable us find length of CL.

Find attached the diagram

Given:

△ABC is an isosceles triangle

KL || AB

AB =8

△BLK and △AKB are also isosceles triangles

From our diagram,

AC = CB (two sides of an isosceles triangles are equal)

∆KCL is similar to ∆ACB

In similar triangles theorem, the ratio of their corresponding sides are equal

KC/AC = LC/BC

Quadrilateral KLAB is an isosceles trapezium:

KL is parallel to AB

∠A = ∠B (opposite base angles are equal)

KA = KL (opposite sides are equal)

The diagonals are congruent

AL ≅ KB

△BLK and △AKL are also isosceles triangles.

BL = AK

∠L = ∠K (opposite base angles are equal)

LK = KL

AL = KB

Also from similar triangles: KL/AB = KC/AC

KL/8 = KC/AC

KC/AC = KL/8

KC/AC = LC/BC

Therefore: KL/8 = LC/BC

Cross multiplying

8CL = KL × CB

CL = (KL × CB)/8