## Thursday, December 25, 2014

### Triangle and Trapezoid Problems, 2

Category: Plane Geometry

"Published in Vacaville, California, USA"

Find the area of the shaded region Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above Photo by Math Principles in Everyday Life

The small triangle is an isosceles triangle because the two adjacent sides are congruent. Same thing with the big triangle. If the bases of two triangles are parallel with the same or common vertex, then two triangles are similar.

Since the two bases of the shaded region is parallel and the opposite sides are congruent, then the figure of the shaded region is an isosceles trapezoid.

The length of the lower base as well as the altitude of an isosceles trapezoid are unknown, then we need to label further the figure first as follows Photo by Math Principles in Everyday Life

The altitude of big triangle bisects the bases of the small and big triangles with their same or common vertex.

By using ratio and proportion since two triangles are similar, the length of the lower base of an isosceles trapezoid is

By Pythagorean Theorem, the altitude of the small triangle is

By using ratio and proportion since two triangles are similar, the altitude of an isosceles trapezoid is

Therefore, the area of the shaded region which is an isosceles trapezoid is

or