https://file.aiquickdraw.com/mathbot/file/1728502440614vsko0vqp.jpg

Let's go through each part of the question.

### 1. **Finding the Zoom Factor (New/Original)**
The zoom factor compares the sides of the two triangles. We take a pair of corresponding sides:

- The base of the original triangle is 20, and the base of the new triangle is 12.
- The height of the original triangle is 15, and the height of the new triangle is 9.

The zoom factor is calculated as:
\[
\text{Zoom Factor} = \frac{\text{New Side}}{\text{Original Side}} = \frac{12}{20} = \frac{3}{5} = 0.6
\]

So, the zoom factor is **0.6**.

### 2. **Finding the Hypotenuse of the New Triangle**
To find the hypotenuse of the new triangle, we use the Pythagorean theorem:

\[
\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2
\]
\[
\text{Hypotenuse}^2 = 12^2 + 9^2 = 144 + 81 = 225
\]
\[
\text{Hypotenuse} = \sqrt{225} = 15
\]

The hypotenuse of the new triangle is **15**.

### 3. **Finding the Area of the New Triangle**
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
\[
\text{Area} = \frac{1}{2} \times 12 \times 9 = \frac{1}{2} \times 108 = 54
\]

So, the area of the new triangle is **54 square units**.

### Summary
1. **Zoom Factor**: 0.6
2. **Hypotenuse of the New Triangle**: 15
3. **Area of the New Triangle**: 54

Would you like more details on any of these steps or have any further questions?

Here are some related questions for further exploration:
1. How do you find the zoom factor if only the hypotenuse lengths are given?
2. How can you confirm if two triangles are similar using their angles?
3. What happens to the area of a triangle when the zoom factor is doubled?
4. How does the zoom factor affect the perimeter of similar triangles?
5. Can two triangles with the same zoom factor have different angles?

**Tip:** Always check if triangles are similar (same shape but different sizes) before calculating zoom factors.

Examine the triangles below: The original triangle has a base of 20 units, a height of 15 units, and an unknown hypotenuse. The new triangle has a base of 12 units, a height of 9 units, and an unknown hypotenuse. The following questions are asked: 1. What is the zoom factor (new/original)? 2. What is the hypotenuse of the new figure? 3. What is the area of the new figure?

This problem explores how to find the zoom factor between two similar triangles, the hypotenuse of the scaled triangle using the Pythagorean theorem, and the area of the new triangle. The base and height of both triangles are provided, and step-by-step calculations are presented.

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