Scale Factor

How to use the meaning of the scale factor of two similar triangles to find the missing side: 2 examples and their solutions.

Similar

Definition

If two triangles are similar,
their shapes are the same
but their sizes are different.

So their corresponding interior angles
are the same
and their corresponding sides
are changed in the same ratio.

[ ~ ] is the similar symbol.

Scale Factor

Definition

That ratio is the scale factor.

The scale factor means
how much the length (sides, height, etc)
has changed.

Example 1

Example

Solution

The given triangles are similar.

See the right triangle.
Side 3 is adjacent to
a circle angle and a plane angle.

See the left triangle.
Side 2 is also adjacent to
a circle angle and a plane angle.

So 2 and 3 sides are the corresponding sides.

2 and 3 are the corresponding sides.
And the scale factor is k.

So 2⋅k = 3.

Divide both sides by 2.

Then k = 3/2.

So the scale factor k is 3/2.

Example 2

Example

Solution

The given triangles are similar.

See the right triangle.
Side x is adjacent to
a circle angle and a square angle.

See the left triangle.
Side 8 is also adjacent to
a circle angle and a square angle.

So x and 8 sides are the corresponding sides.

8 and x are the corresponding sides.
And the scale factor is 2.

So x = 8⋅2.

8⋅2 = 16

So 16 is the answer.