Question

your turn

1. to find the distance d across a stream, levi located points as shown in the figure. use the given information to find d.

evaluate independent practice
lesson 12.3 homework
complete problems 1-6 below for independent practice.

1. if \\(\delta fgh \sim \delta tsr\\), find the value of \\(gh\\).
2. if \\(\delta fgh \sim \delta tuh\\), find the value of \\(ht\\).
3. if \\(\delta wvt \sim \delta rst\\), find the value of \\(vt\\).
4. \\(\delta klm \sim \delta kcd\\) are similar. solve for \\(x\\).
5. \\(\delta efg \sim \delta ekl\\) are similar. solve for \\(x\\).
6. \\(\delta klm \sim \delta kqp\\) are similar. solve for \\(x\\).

Explanation:

Identify similar triangles and set up proportion

Using the Geometric Similarity Applications knowledge point, we analyze the "Your Turn" problem at the top of the page.
The diagram shows two right triangles sharing a vertical angle at vertex \(C\).
Specifically, \(\triangle ABC \sim \triangle DEC\) because:

• \(\angle B \cong \angle E\) (both are right angles, \(90^\circ\))
• \(\angle ACB \cong \angle DCE\) (vertical angles)

Thus, by AA similarity, the triangles are similar.

Write the ratio of corresponding sides

Using the Corresponding Sides of Similar Figures knowledge point, we set up the ratio of corresponding sides:

$$ \frac{AB}{DE} = \frac{BC}{EC} $$

From the given figure, we identify the lengths:

• \(AB = d\)
• \(DE = 12\text{ m}\)
• \(BC = 12\text{ m}\)
• \(EC = 6\text{ m}\)

Substituting these values into the proportion:

$$ \frac{d}{12} = \frac{12}{6} $$

Solve for the unknown distance d

Using the Solving Linear Equations knowledge point

$$
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$$

Answer:

The distance across the stream is \(d = 24\text{ m}\).