To solve for the length $LN$ using the Pythagorean Theorem, we can break this into two steps because two right triangles are involved.

Given:

• Triangle $KLM$ is a right triangle with:

  • $KL = 25$ (hypotenuse)
  • $KM = 17$ (one leg)
  • We need to find $LM$ (the other leg).

• Triangle $LMN$ is another right triangle with:

  • $LM$ (hypotenuse of this triangle) will be calculated in the first part.
  • $MN = 8$ (one leg of the second triangle)
  • We need to find $LN$ (the other leg).

Step 1: Find $LM$ using triangle $KLM$

Apply the Pythagorean theorem to triangle $KLM$: $KL^2 = KM^2 + LM^2$ $25^2 = 17^2 + LM^2$ $625 = 289 + LM^2$ $LM^2 = 625 - 289 = 336$ $LM = \sqrt{336} \approx 18.33$

Step 2: Find $LN$ using triangle $LMN$

Now, apply the Pythagorean theorem to triangle $LMN$: $LM^2 = LN^2 + MN^2$ $18.33^2 = LN^2 + 8^2$ $336 = LN^2 + 64$ $LN^2 = 336 - 64 = 272$ $LN = \sqrt{272} \approx 16.49$

Final Answer:

The length of $LN$ is approximately $16$ when rounded to the nearest whole number.

Would you like any further clarification or details on this?

Here are 5 related questions to expand on this topic:

1. How does the Pythagorean theorem apply to other shapes besides right triangles?
2. What would happen if the triangle was not a right triangle?
3. How can the Pythagorean theorem be used in real-world applications?
4. What are other methods to calculate the length of a triangle's side besides the Pythagorean theorem?
5. How do you calculate the distance between two points in 3D space using a similar method?

Tip: Always double-check your calculations when dealing with square roots and rounding, especially in applied problems.