https://file.aiquickdraw.com/mathbot/file/172850223411866cg4l0u.jpg

Let's break down the question and solve it step by step.

### 1. ZOOM FACTOR:
We are given two similar triangles, and we need to find the zoom factor (or scale factor) from the original triangle to the new triangle. The scale factor is determined by comparing the lengths of corresponding sides.

- **Original Figure's base (long leg):** 20
- **New Figure's base (long leg):** 8

The zoom factor is the ratio of the new triangle’s base to the original triangle’s base:

\[
\text{Zoom Factor} = \frac{\text{New Figure's base}}{\text{Original Figure's base}} = \frac{8}{20} = \frac{2}{5}
\]

So, the **Zoom Factor** is \( \frac{2}{5} \) or 0.4.

### 2. Hypotenuse of the new figure:
We can calculate the hypotenuse of the new triangle using the Pythagorean theorem:

\[
\text{Hypotenuse}^2 = (\text{Base})^2 + (\text{Height})^2
\]

For the **new figure**, the base is 8, and the height is 6:

\[
\text{Hypotenuse}^2 = 8^2 + 6^2 = 64 + 36 = 100
\]

Taking the square root of both sides:

\[
\text{Hypotenuse} = \sqrt{100} = 10
\]

Thus, the hypotenuse of the new figure is **10**.

### 3. Area of the new figure:
The area of a triangle is given by:

\[
\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]

For the **new figure**, the base is 8, and the height is 6:

\[
\text{Area} = \frac{1}{2} \times 8 \times 6 = \frac{1}{2} \times 48 = 24
\]

Thus, the area of the new figure is **24** square units.

### Summary of Answers:
1. **Zoom Factor:** 0.4 or \( \frac{2}{5} \)
2. **Hypotenuse of the new figure:** 10
3. **Area of the new figure:** 24

Would you like more details or have any questions about these calculations?

Here are 5 related questions to expand your understanding:
1. How would the zoom factor change if the new triangle’s base was 10 instead of 8?
2. Can you use a different pair of corresponding sides to calculate the zoom factor?
3. What happens to the area of the triangle if the zoom factor is doubled?
4. How does the hypotenuse change with the zoom factor?
5. How can the zoom factor help in calculating the side lengths of other similar triangles?

**Tip:** In similar triangles, the ratio of corresponding sides is always the same, and the area scales by the square of the zoom factor.

Examine the triangles below. The ZOOM FACTOR (New/Original) for the above triangles is ______. The hypotenuse of the new figure is ______. The area of the new figure is ______.

Learn how to find the zoom factor, hypotenuse, and area of a triangle based on its similar counterpart. The problem uses the Pythagorean theorem and area formula to calculate the hypotenuse and area. Suitable for Grades 7-10 geometry students.

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