Similarity of Sides in a Right Triangle

Similarity of Sides in a Right Triangle

How to solve the similarity of sides in a right triangle problems: examples, and their solutions.

Example 1

Find the value of x. The measures of the right triangle's base: 16 + 9. The height of the right triangle: x.

From the given right triangle,
there are 3 similar right triangles available:
the left one, the right one, and the whole triangle.

Let's use the left one and the right one.

Draw the left triangle and the right triangle.
Draw the right triangles
to show their similarity easily.

These two triangles have two congruent angles.
So, by the AA similarity,
these two triangles are similar.

Write a proportion
using their corresponding sides:
x / 9 = 16 / x.

You don't have to put ± sign,
because x > 0.

Simplifying a radical (part 1)

Example 2

Find the values of x and y. The measures of the right triangle's base: y + 1. The measure of the longer leg of the right triangle: x. The height of the right triangle: square root 3.

Draw the left triangle, the right triangle,
and the whole triangle.

Draw the right triangles
to show their similarities easily.

From the purple and blue triangles,
write a proportion
using their corresponding sides:
1 / √3 = √3 / y.

From the blue and green triangles,
write a proportion
using their corresponding sides:
x / (y + 1) = y / x.

Solve the first proportion.
(It has only one variable: y.)
Then y = 3.

Put y = 3 into the second proportion.
Then x = 2√3.

Simplifying a radical (part 1)