Question

your turn
solve for the missing variable, angle measure, or side length.

1. find the value of (x). you may assume triangle abc is similar to triangle stu.
2. what must the missing length be for the two figures to be similar?

use the diagram, in which (delta abe sim delta acd).

1. find the value of (x).

Explanation:

Set up similarity ratios for Question 1

Using the Similarity Statements and Corresponding Sides of Similar Figures knowledge points:
Given \(\triangle ABC \sim \triangle STU\), the ratio of corresponding sides must be equal:

$$ \frac{AB}{ST} = \frac{AC}{SU} $$

Substitute the given values \(AB = 63\), \(AC = 35\), \(ST = 9\), and \(SU = 5\):

$$ \frac{63}{9} = \frac{35}{5} = 7 $$

The scale factor is \(7\). Now, match the corresponding angles using the Corresponding Angles of Similar Figures knowledge point:

$$ m\angle A = m\angle S \implies 4x = 100 $$

Solve for x in Question 1

Using the Solving Linear Equations knowledge point:

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Set up similarity ratios for Question 2

Using the Corresponding Sides of Similar Figures knowledge point:
Identify corresponding sides of the two similar quadrilaterals:

• The side of length \(6\) corresponds to the side of length \(9\).
• The side of length \(12\) corresponds to the side of length marked with \(?\).

Set up the proportion:

$$ \frac{6}{9} = \frac{12}{?} $$

Solve for the missing length in Question 2

Using the Solving Linear Equations knowledge point:

$$
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Solve for x in Question 3

Using the Corresponding Angles of Similar Figures and Solving Linear Equations knowledge points:
Given \(\triangle ABE \sim \triangle ACD\), the corresponding angles must be equal:

$$ m\angle ABE = m\angle ACD $$

From the diagram, \(m\angle ABE = (3x + 14)^\circ\) and \(m\angle ACD = 50^\circ\):

$$
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Answer:

Question 1

\(x = 25\)

Question 2

The missing length is \(18\)

Question 3

\(x = 12\)