Question

In the figures shown, both triangles are right triangles.



In Figure 1, the area of square A is 9 square units, the area of square B is 16 square units, and the area of square C is 25 square units.

In Figure 2, the area of square D is 36 square units and the area of square E is 64 square units.

What is the area of square F in Figure 2?
A.
110 square units
B.
100 square units
C.
80 square units
D.
90 square unit

Answers

GPT-4o mini
To find the area of square F in Figure 2, we first need to understand the relationship between the squares in the context of right triangles.

In Figure 1, let’s denote:
- The area of square A (one leg of the right triangle) = 9 square units, which means the side length is \( \sqrt{9} = 3 \) units.
- The area of square B (the other leg) = 16 square units, which gives a side length of \( \sqrt{16} = 4 \) units.
- The area of square C (the hypotenuse) = 25 square units, with a side length of \( \sqrt{25} = 5 \) units.

Since these are right triangles, we can confirm that:
\[
3^2 + 4^2 = 9 + 16 = 25 = 5^2
\]
which is consistent with the Pythagorean theorem.

In Figure 2, we have:
- The area of square D (one leg of the right triangle) = 36 square units, giving a side length of \( \sqrt{36} = 6 \) units.
- The area of square E (the other leg) = 64 square units, with a side length of \( \sqrt{64} = 8 \) units.

We can again check for the hypotenuse calculated from these values:
\[
6^2 + 8^2 = 36 + 64 = 100
\]
Thus, the area of square F, which represents the hypotenuse, is \( 100 \) square units.

Hence, the area of square F in Figure 2 is:
\[
\boxed{100} \text{ square units.}
\]

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