Similar Triangles

How to find the side from the given similar triangles: 3 examples and their solutions.

Similar Triangles: in a Right Triangle

Example 1

Example

Solution

For the given two triangles,
two angles of each triangle are congruent.

So, by the AA similarity,
these two triangles are similar.

Write [ ~ ] between the triangles.

These two triangles are similar.

Then their sides are proportional.

So 4/6 = 5/x.

4/6 = 2/3

Solve the proportion.

Then 2x = 5⋅3.

5⋅3 = 15

Divide both sides by 2.

Then x = 15/2.

So x = 15/2.

Example 2

Example

Solution

The brown dot angles are
alternate interior angles in parallel lines.

So the brown dot angles are congruent.

The brown angles are vertical angles.

So the brown angles are also congruent.

Let's use these two triangles.

Draw the top triangle.

Make the shape of the top triangle
the same as the bottom triangle.

Draw the bottom triangle.

For these two triangles,
two angles of each triangle are congruent.

So, by the AA similarity,
these two triangles are similar.

Write [ ~ ] between the triangles.

These two triangles are similar.

Then their sides are proportional.

So (3x + 2)/16 = 7/14.

7/14 = 1/2

Solve the proportion.

Then 2(3x + 2) = 16.

Divide both sides by 2.

Then 3x + 2 = 8.

Move +2 to the right side.

Then 3x = 6.

Divide both sides by 3.

Then x = 2.

So x = 2.

Example 3

Example

Solution

From the given figure,
find the similar triangles.

Draw the little triangle.

Draw the whole triangle.

Make the shape of the whole triangle
the same as the little triangle.

The plane angles are the same angle.

So the plane angles are congruent.

For these two triangles,
two angles of each triangle
(plane angle, right angle)
are congruent.

So, by the AA similarity,
these two triangles are similar.

Write [ ~ ] between the triangles.

These two triangles are similar.

Then their sides are proportional.

So 3/7 = 5/(5 + x).

Solve the proportion.

Then 3(5 + x) = 5⋅7.

3(5 + x) = 15 + 3x

Multiply a Monomial and a Polynomial

5⋅7 = 35

Move 15 to the right side.

Then 3x = 20.

Divide both sides by 3.

Then x = 20/3.

So x = 20/3.