A Geometry problem involving triangles

by Soumya Das
(India)

ACEF and ACDB are rectangles. BDEF is a square and the length of each side is 1 unit. If BG has a length of a, then what will be the length of AB?

Solution

First, show that triangle ABG and triangle AFE are similar.

Angle A is the same for triangle ABG and triangle AFE

Similarly, angle F is the same for triangle ABG and triangle AFE


Since triangle ABG and triangle AFE have two angles that have the same measure, the last angle for these two triangles must be the same.

Therefore, triangle ABG and triangle AFE are similar.


Since triangle ABG and triangle AFE are similar, we can use proportion to find ? or the length of AB.

AB/a = (AB + 1)/1

AB × 1 = a × AB + a × 1

AB = a(AB) + a

AB - a(AB) = a

AB(1-a) = a

AB(1-a)/(1-a) = a/1-a

AB = a/1-a