https://file.aiquickdraw.com/mathbot/file/1724647619152vlwouqyd.jpg

The sum of the squares of the lengths of the legs of the triangle is ______.
The square of the length of the hypotenuse of the triangle is _________.

The image contains the following problem:

**The sum of the squares of the lengths of the legs of the triangle is \_\_\_\_\_\_\_.  
The square of the length of the hypotenuse of the triangle is \_\_\_\_\_\_\_.**

Given options are:
- A. 121, 121
- B. 169, 169
- C. 100, 100

This problem refers to the Pythagorean theorem, which states:

\[
a^2 + b^2 = c^2
\]

where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. The problem is asking for both the sum of the squares of the legs and the square of the hypotenuse, which are the same based on this equation.

Therefore, you should select the option where both numbers are equal. 

Here are the options analyzed:

- **Option A**: 121, 121  
- **Option B**: 169, 169  
- **Option C**: 100, 100  

Without additional context or the lengths of the sides, the answer depends on which pair of equal numbers correctly applies to the triangle in question. However, from the image provided, it looks like a general application of the theorem without needing specific side lengths.

If you know the specific lengths of the legs and the hypotenuse, please share them for a more precise evaluation.

Do you want any details, or do you have further questions? Here are five related questions that might interest you:

1. How do you derive the Pythagorean theorem?
2. What are the practical applications of the Pythagorean theorem?
3. How would the theorem apply to non-right triangles?
4. What are some alternative ways to prove the Pythagorean theorem?
5. Can you explain how the theorem relates to distance in coordinate geometry?

**Tip:** Remember, the Pythagorean theorem only applies to right-angled triangles. Ensure you're working with such a triangle before applying the formula!

Explore how to solve problems using the Pythagorean theorem, focusing on calculating the sum of the squares of the legs and the square of the hypotenuse. This problem involves applying the Pythagorean theorem to find equal values for both quantities. Suitable for students in Grades 7-9.

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