# Similar Triangles: in a Right Triangle

How to find the similar triangles in a right triangle to find the side of the right triangle: 2 examples and their solutions.

## Example

Draw a plane angle and a dot angle like this.

Then m∠[plane] + m∠[dot] = 90.

Complementary Angles

See the left triangle.

Its interior angles are

a plane angle and a right angle.

So the top left angle is

a dot angle.

See the right triangle.

Its interior angles are

a right angle and a dot angle.

So the top right angle is

a plane angle.

From this right triangle,

let's find the similar triangles.

Draw the right side triangle.

Draw the left triangle.

Draw the shape of the left triangle

the same as the drawn triangle.

For these two triangles,

three angles of each triangle are congruent.

So these two triangles are similar.

Write [ ~ ] between the triangles.

These two triangles are similar.

Then their sides are proportional.

So 9/x = x/16.

Solve the proportion.

Then x^{2} = 9⋅16.

9 = 3^{2}

16 = 4^{2}

Square root both sides.

Then x = 3⋅4.

x is the lenght of a side.

So x is plus.

So you don't have to write ± sign.

Quadratic Equation: Square Root

3⋅4 = 12

So x = 12.

## Example

Draw a plane angle and a dot angle like this.

Then m∠[plane] + m∠[dot] = 90.

See the left triangle.

Its interior angles are

a plane angle and a right angle.

So the top left angle is

a dot angle.

See the right triangle.

Its interior angles are

a right angle and a dot angle.

So the top right angle is

a plane angle.

From this right triangle,

let's find the similar triangles.

Draw the right side triangle.

Draw the left triangle.

Draw the shape of the left triangle

the same as the drawn triangle.

Draw the whole triangle.

Draw the shape of the whole triangle

the same as the drawn triangles.

For these three triangles,

three angles of each triangle are congruent.

So these three triangles are similar.

Write [ ~ ] between the triangles.

Finding y seems to be simpler.

Let's find y first.

See the left and middle triangles.

These two triangles are similar.

Then their sides are proportional.

So √3/y = 1/√3.

Solve the proportion.

Then y = √3⋅√3 = 3.

Put y = 3

into the y variables.

y = 3

y + 1 = 3 + 1 = 4

Next, see the middle and right triangles.

These two triangles are similar.

Then their sides are proportional.

So x/4 = 3/x.

Solve the proportion.

Then x^{2} = 3⋅4.

4 = 2^{2}

x^{2} = 3⋅2^{2}

Square root both sides.

Then x = √3⋅2^{2}.

x is the lenght of a side.

So x is plus.

So you don't have to write ± sign.

√3⋅2^{2} = 2√3

Simplify a Radical

y = 3

x = 2√3

So write the answer:

x = 2√3, y = 3.

So x = 2√3 and y = 3.