Question

Kierra draws a triangle with a base of 10 cm and a height of 15 cm. She dilates the triangle using a scale factor of f
• Then, Kierra finds the difference between the two areas by subtracting.
How much greater is the area of the original triangle than the area of the dilated triangle? (1 point)
• 48 cm?
• 54 cm?
• 27 cm?
O 75 cm?

Answer 1

To solve the problem, we first calculate the area of the original triangle.

The formula for the area of a triangle is given by:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

1. **Calculate the area of the original triangle:**
- Base = 10 cm
- Height = 15 cm

\[
\text{Area}_{\text{original}} = \frac{1}{2} \times 10 \times 15 = 75 \text{ cm}^2
\]

2. **Calculate the area of the dilated triangle:**
Let's denote the scale factor as \( f \). When a triangle is dilated by a scale factor \( f \), both the base and height are multiplied by \( f \). Therefore, the new base and height are:

- Base of the dilated triangle = \( 10f \) cm
- Height of the dilated triangle = \( 15f \) cm

The area of the dilated triangle is:

\[
\text{Area}_{\text{dilated}} = \frac{1}{2} \times (10f) \times (15f) = \frac{1}{2} \times 150f^2 = 75f^2 \text{ cm}^2
\]

3. **Find the difference between the two areas:**

Now, we need to find the difference between the area of the original triangle and the area of the dilated triangle:

\[
\text{Difference} = \text{Area}_{\text{original}} - \text{Area}_{\text{dilated}} = 75 \text{ cm}^2 - 75f^2 \text{ cm}^2
\]

\[
\text{Difference} = 75(1 - f^2) \text{ cm}^2
\]

4. **Determine the answer based on the scale factor \( f \):**

If \( f \) is a positive number less than 1, \( 1 - f^2 \) will be positive, meaning the area of the original triangle will be greater than that of the dilated triangle. However, we do not have a specific value for \( f \) to calculate the exact difference.

None of the choices (48 cm², 54 cm², 27 cm², or 75 cm²) can be determined without knowing the value of \( f \).

Assuming Kierra dilated the triangle with a scale factor that is appropriate to any of the choices, we can substitute \( f \) accordingly to find possible values for the difference.

- If \( f = 1 \), then the difference is \( 75(1 - 1^2) = 0 \) cm² (not an option).
- Assuming \( f \) close to \( 0.6 \) gives:

\[
f^2 = 0.36
\]
\[
\text{Difference} = 75(1 - 0.36) = 75(0.64) = 48 \text{ cm}^2
\]
Which is one of the provided options.

Thus, the answer is:
**48 cm²**.