Topics
6. Polygons
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United States of AmericaVirginia
Grade 7

6.03 Find unknown measures in similar figures

Lesson
Practice

Find unknown measures in similar figures

Recall the side lengths of similar shapes are in the same ratio or proportion.

If two figures are similar, we can use the definition of similarity to find unknown measures.

2 triangles: triangle ABC and triangle DEF. Triangle ABC have angles 78 degrees, 46 degrees, and 56 degrees. Triangle DEF have angles 78 degrees, x degrees, and 56 degrees.

Consider these triangles where we do not know the measure of \angle E, but we are told:

\triangle ABC \sim \triangle DEF

We know similar figures have congruent corresponding angles. The similarity statement, tells us \angle B corresponds to \angle E.

If angles are congruent, they have the same degree measure, so x=46\degree.

Two quadrilaterals: HJKM and ABCD. Quadrilateral HJKM have side lengths of 3 inches, x inches, 3.5 inches, and 4 inches. Quadrilateral ABCD have side lengths of 8.4 inches, 7 inches, 9.8 inches, and 11.2 inches.

If these figures are similar, find the value of x.

It is difficult to identify the scale factor when one side is not a clear multiple of its corresponding side. In situations like this we can write a proportion.

A proportion is an equation involving the ratios of corresponding sides. We need to make sure to use the pair of corresponding sides that includes the unknown side length x for one side of the equation. For the other side, we can choose any other corresponding pair as long as we know both side lengths.

Let's use sides \overline{JK} (the unknown) and \overline{BC} for the left side and sides \overline{KM} and \overline{CD} for the right side.

\displaystyle \dfrac{JK}{BC}\displaystyle =\displaystyle \dfrac{KM}{CD}Write the proportion of the side lengths
\displaystyle \dfrac{x}{7}\displaystyle =\displaystyle \dfrac{3.5}{9.8}Substitute the known values
\displaystyle x \cdot 9.8\displaystyle =\displaystyle 7 \cdot 3.5Means extremes property
\displaystyle 9.8x\displaystyle =\displaystyle 24.5Evaluate the multiplication
\displaystyle \dfrac{9.8x}{9.8}\displaystyle =\displaystyle \dfrac{24.5}{9.8}Divide both sides by 9.8
\displaystyle x\displaystyle =\displaystyle 2.5 Evaluate the division

So the length of \overline{JK} is 2.5 inches.

We can confirm this by finding the scale factor which is the ratio between the corresponding sides. The ratio CD:KM=9.8:3.5 or \dfrac{9.8}{3.5}. Simplifying that fraction we get a scale factor of 2.8.

To get from the side with length x=2.5 to its corresponding side with length 7 we can multiply 2.5 \cdot 2.8=7. This confirms the length of side \overline{JK}.

Examples

Example 1

Given that the two quadrilaterals below are similar, what is the value of x?

Two quadrilaterals: ABCD and PQRS. Quadrilateral ABCD have side lengths of 7.5, 6, 4, 3 units and angles of 105 degrees and 54 degrees. Side PS has a side length of 2 units and Quadrilateral PQRS have angles of 54 degrees and x degrees.
Worked Solution
Create a strategy

Corresponding angles are congruent in similar figures. So we need to find the measure of the angle that corresponds with \angle S.

Apply the idea

\angle S corresponds with \angle D so \angle S \cong \angle D.

Congruent angles have the same degree measure.

x= 105

Reflect and check

Keep in mind the value of x is 105, but m\angle S = 105\degree.

Example 2

Given that the two quadrilaterals below are similar, find the value of w.

Two quadrilaterals: EFGH and ABCD. Side EF measures 16 units, side FG measures 8 units, side AB measures w units, and side BC measures 18 units.
Worked Solution
Create a strategy

Write the proportion between the corresponding sides to find w. The side \overline{EF} corresponds to the side \overline{AB} which has the length w.

Apply the idea
\displaystyle \dfrac{16}{w}\displaystyle =\displaystyle \dfrac{8}{18}Write the proportion for the given sides
\displaystyle \dfrac{16}{w}\displaystyle =\displaystyle \dfrac{4}{9}Simplify the fraction
\displaystyle 16 \cdot 9\displaystyle =\displaystyle w\cdot4Means extremes property
\displaystyle 144\displaystyle =\displaystyle 4wEvaluate the multiplication
\displaystyle \dfrac{144}{4}\displaystyle =\displaystyle \dfrac{4w}{4}Divide both sides by 4
\displaystyle 36\displaystyle =\displaystyle wEvaluate the division
Reflect and check

We can confirm this by finding the scale factor which is the ratio between the corresponding sides.

The ratio BC : FG = 18:8 or \dfrac{18}{8}. Simplifying that fraction we get a scale factor of \dfrac{9}{4}. This is greater than 1 which makes sense because the shape was enlarged.

To get from length EF=16 to its corresponding side length AB=w we can multiply {16 \cdot \dfrac{9}{4} = \dfrac{144}{4}=36}. This confirms the length of side \overline{AB}.

Example 3

A 4.5\text{ m} high flagpole casts a shadow of 4.4\text{ m}. At the same time, the shadow of a nearby building falls at the same point (S). The shadow cast by the building creates a pair of similar triangles and measures 8.8\text{ m}. Find h, the height of the building, using a proportion statement.

This image shows a flagpole and a building casting shadows. Ask your teacher for more information.
Worked Solution
Create a strategy

Write a proportion to represent the ratios of the corresponding sides of the similar triangles.

Apply the idea
\displaystyle \dfrac{h}{4.5}\displaystyle =\displaystyle \dfrac{8.8}{4.4}Write a proportion with the corresponding sides
\displaystyle \dfrac{h}{4.5}\displaystyle =\displaystyle 2Simplify the fraction
\displaystyle 4.5\cdot\dfrac{h}{4.5} \displaystyle =\displaystyle 2\cdot 4.5Multiply both sides by 4.5
\displaystyle h \displaystyle =\displaystyle 9 \text{ m}Evaluate
Idea summary

If two figures are similar their:

  • Corresponding angles are congruent
  • Corresponding sides are proportional

Unknown side lengths can be found by writing a proportion with the corresponding sides and solving using the means extremes property.

Outcomes

7.MG.2

The student will solve problems and justify relationships of similarity using proportional reasoning.

7.MG.2d

Write proportions to express the relationships between the lengths of corresponding sides of similar quadrilaterals and triangles.

7.MG.2f

Solve a proportion to determine a missing side length of similar quadrilaterals or triangles.

7.MG.2g

Given angle measures in a quadrilateral or triangle, determine unknown angle measures in a similar quadrilateral or triangle.

7.MG.2h

Apply proportional reasoning to solve problems in context including scale drawings. Scale factors shall have denominators no greater than 12 and decimals no less than tenths.

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