# 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

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

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.