Lesson 9

Side Length Quotients in Similar Triangles

Let’s find missing side lengths in triangles.

Triangle $A$ has side lengths 2, 3, and 4. Triangle $B$ has side lengths 4, 5, and 6. Is Triangle $A$ similar to Triangle $B$ ?

Triangle $ABC$ is similar to triangles $DEF$ , $GHI$ , and $JKL$ . The scale factors for the dilations that show triangle $ABC$ is similar to each triangle are in the table.

Triangle A, B C. Side A, B length 4, side B C length 7, side C A, length 5.

Find the side lengths of triangles $DEF$ , $GHI$ , and $JKL$ . Record them in the table.

triangle	scale factor	length of short side	length of medium side	length of long side
$ABC$	1	4	5	7
$DEF$	2
$GHI$	3
$JKL$	$\frac{1}{2}$

Your teacher will assign you one of the three columns. For all four triangles, find the quotient of the triangle side lengths assigned to you and record it in the table. What do you notice about the quotients?

triangle	(long side) $\div$ (short side)	(long side) $\div$ (medium side)	(medium side) $\div$ (short side)
$ABC$	$\frac{7}{4}$ or 1.75
$DEF$
$GHI$
$JKL$

Compare your results with your partners’ and complete your table.

Are you ready for more?

Triangles $ABC$ and $DEF$ are similar. Explain why $\frac{AB}{BC} = \frac{DE}{EF}$ .

Two triangles. First A, B C. Second D E F.

Triangles $ABC$ , $EFD$ , and $GHI$ are all similar. The side lengths of the triangles all have the same units. Find the unknown side lengths.

If two polygons are similar, then the side lengths in one polygon are multiplied by the same scale factor to give the corresponding side lengths in the other polygon.

For these triangles the scale factor is 2:

Two triangles. First, A, B C. Length A, B, 5, length B C, 4, length C A, 3. Second, A, prime B prime C prime. Length A, prime B prime, 10, B prime C prime 8, C prime A, prime.

Here is a table that shows relationships between the short and medium length sides of the small and large triangle.

	small triangle	large triangle
medium side	4	8
short side	3	6
(medium side) $\div$ (short side)	$\frac{4}{3}$	$\frac{8}{6} = \frac{4}{3}$

The lengths of the medium side and the short side are in a ratio of $4:3$ . This means that the medium side in each triangle is $\frac43$ as long as the short side. This is true for all similar polygons; the ratio between two sides in one polygon is the same as the ratio of the corresponding sides in a similar polygon.

We can use these facts to calculate missing lengths in similar polygons. For example, triangles $A’B’C’$ and $ABC$ shown here are similar. Let's find the length of segment $B’C’$ .

In triangle $ABC$ , side $BC$ is twice as long as side $AB$ , so this must be true for any triangle that is similar to triangle $ABC$ . Since $A'B'$ is 1.2 units long and $2\boldcdot 1.2 = 2.4$ , the length of side $B’C’$ is 2.4 units.

Two triangles. First, A, B C. Length A, B, 3, length B C,6. Second, A, prime B prime C prime. Length A, prime B prime, 1 point 2.

Video VLS G8U2V2 Similarity (Lesson 6–9) available at https://player.vimeo.com/video/457854496.

similar

Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.

In this figure, triangle $ABC$ is similar to triangle $DEF$ .

If $ABC$ is rotated around point $B$ and then dilated with center point $O$ , then it will fit exactly over $DEF$ . This means that they are similar.

Lesson 9

9.1: Two-three-four and Four-five-six

9.2: Quotients of Sides Within Similar Triangles

9.3: Using Side Quotients to Find Side Lengths of Similar Triangles

Summary

Video Summary

Glossary Entries