https://file.aiquickdraw.com/mathbot/file/1729609345495qoylnxki.jpg

The given image shows a right triangle \( \triangle EHF \) with:

- \( EH = 9 \, \text{in} \)
- Hypotenuse \( HF = 15 \, \text{in} \)
- \( EF \) is the unknown length we need to find.

This is a classic right triangle, so we can use the **Pythagorean theorem**:

\[
EH^2 + EF^2 = HF^2
\]

Substitute the known values:

\[
9^2 + EF^2 = 15^2
\]
\[
81 + EF^2 = 225
\]
\[
EF^2 = 225 - 81
\]
\[
EF^2 = 144
\]
\[
EF = \sqrt{144} = 12 \, \text{in}
\]

So, \( EF = 12 \, \text{in} \).

### Now for the area of rectangle \( EFGH \):

The area of the rectangle is calculated using:

\[
\text{Area} = EH \times EF
\]

Substitute the values:

\[
\text{Area} = 9 \times 12 = 108 \, \text{square inches}
\]

### Summary:
- \( EF = 12 \, \text{in} \)
- Area of rectangle \( EFGH \) = 108 square inches.

Would you like more details on any part of this process?

Here are 5 related questions to explore:
1. How is the Pythagorean theorem used for other triangle types?
2. What other ways can we calculate the area of polygons?
3. How do you apply the Pythagorean theorem in 3D geometry?
4. Can you use the Pythagorean theorem in non-Euclidean geometry?
5. What are the properties of right triangles and their applications?

**Tip:** Whenever dealing with right triangles, check if it fits common Pythagorean triples like \(3-4-5\) or \(9-12-15\). This can save time!

Find the length of EF and the area of rectangle EFGH, given EH = 9 inches, HF = 15 inches, and EF as the unknown side.

This problem involves calculating the unknown side EF in a right triangle using the Pythagorean theorem and finding the area of rectangle EFGH. With EH = 9 inches and HF = 15 inches, follow step-by-step to solve for EF and the area.

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