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△JKL∼△PKQ , and KP¯¯¯¯¯¯¯¯=14 m, JK¯¯¯¯¯¯¯¯=56 m, and KQ¯¯¯¯¯¯¯¯=21 m. What is the length of KL¯¯¯¯¯¯¯¯ ?

The given problem involves two similar triangles, \( \triangle JKL \) and \( \triangle PKQ \), as indicated by \( \triangle JKL \sim \triangle PKQ \). We are given the following lengths:

- \( KP = 14 \, \text{m} \)
- \( JK = 56 \, \text{m} \)
- \( KQ = 21 \, \text{m} \)

We need to find the length of \( KL \).

### Step 1: Use of Similar Triangles
Since \( \triangle JKL \sim \triangle PKQ \), the corresponding sides of the triangles are proportional. Therefore, we can set up a proportion between the corresponding sides:

\[
\frac{JK}{KP} = \frac{KL}{KQ}
\]

Substitute the known values:

\[
\frac{56}{14} = \frac{KL}{21}
\]

### Step 2: Simplify and Solve for \( KL \)
Simplifying the left-hand side:

\[
\frac{56}{14} = 4
\]

Now, solve for \( KL \):

\[
4 = \frac{KL}{21}
\]

Multiply both sides by 21 to isolate \( KL \):

\[
KL = 4 \times 21 = 84 \, \text{m}
\]

Thus, the length of \( KL \) is \( 84 \, \text{m} \).

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### Related Questions:
1. How can we use similar triangles to find unknown angles?
2. What are the properties of similar triangles in terms of angles and sides?
3. How do we prove that two triangles are similar?
4. What is the significance of proportional sides in similar triangles?
5. How does the Pythagorean theorem relate to right triangles in similarity problems?

### Tip:
When dealing with similar triangles, always remember that their corresponding sides are proportional, and their corresponding angles are equal. This property helps in solving many geometry problems involving ratios.

△JKL∼△PKQ, and KP = 14 m, JK = 56 m, and KQ = 21 m. What is the length of KL?

Learn how to find the length of KL using the properties of similar triangles △JKL and △PKQ. The problem involves proportional sides and is suitable for students in Grades 8-10.

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