Lesson 9Side Length Quotients in Similar Triangles

Learning Goal

Let’s find missing side lengths in triangles.

Learning Targets

I can decide if two triangles are similar by looking at quotients of lengths of corresponding sides.
I can find missing side lengths in a pair of similar triangles using quotients of side lengths.

Lesson Terms

similar

Warm Up: Two-three-four and Four-five-six

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

Activity 1: Quotients of Sides Within Similar Triangles

Your teacher will assign you one of the three columns in the second table.

Triangle $A B C$ is similar to triangles $D E F$ , $G H I$ , and $J K L$ . The scale factors for the dilations that show triangle $A B C$ is similar to each triangle are in the table.

A triangle ABC with side length 4, 5, 7.

Find the side lengths of triangles $D E F$ , $G H I$ , and $J K L$ . Record them in the table.
triangle
scale factor
length of short side
length of medium side
length of long side
$A B C$
$1$
$4$
$5$
$7$
$D E F$
$2$
$G H I$
$3$
$J K L$
$\frac{1}{2}$
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)
$A B C$
$\frac{7}{4}$ or 1.75
$D E F$
$G H I$
$J K L$
Compare your results with your partners’ and complete your table.

triangle	scale factor	length of short side	length of medium side	length of long side
$A B C$	$1$	$4$	$5$	$7$
$D E F$	$2$
$G H I$	$3$
$J K L$	$\frac{1}{2}$

Are you ready for more?

Triangles $A B C$ and $D E F$ are similar. Explain why $\frac{A B}{B C} = \frac{D E}{E F}$ .

Activity 2: Using Side Quotients to Find Side Lengths of Similar Triangles

Triangles $A B C$ , $E F D$ , and $G H I$ are all similar. The side lengths of the triangles all have the same units. Find the unknown side lengths.

Triangle ABC with side lengths: AB=c, AC and BC=4. Triangle DEF with side lengths: DE=d, DF=e, EF=5. Triangle GHI with side lengths: GH=six-fifths, GI=h, HI=twelve-fifths

Lesson Summary

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:

Triangle ABC with side lengths 3, 4, 5. Larger triangle A'B'C' with side lengths 6, 8, 10.

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 $\frac{4}{3}$ 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 $A B C$ shown here are similar. Let’s find the length of segment $B^{'} C^{'}$ .

Triangle ABC with side AB length 3 and side BC length 6. Smaller triangle A'B'C' with AB side length 1.2.

In triangle $A B C$ , side $B C$ is twice as long as side $A B$ , so this must be true for any triangle that is similar to triangle $A B C$ . Since $A^{'} B^{'}$ is 1.2 units long and $2 \cdot 1.2 = 2.4$ , the length of side $B^{'} C^{'}$ is 2.4 units.

Lesson 9Side Length Quotients in Similar Triangles

Learning Goal

Learning Targets

Lesson Terms

Warm Up: Two-three-four and Four-five-six

Problem 1

Activity 1: Quotients of Sides Within Similar Triangles

Problem 1

Are you ready for more?

Problem 1

Activity 2: Using Side Quotients to Find Side Lengths of Similar Triangles

Problem 1

Lesson Summary