Use the image to answer the question.

An illustration shows a triangle with a base labeled 8 inches. A line runs through the center of the triangle from the apex to the base. This perpendicular height is labeled 5 inches.

Crystal cuts a piece of wood into the shape of a triangle. The height of the triangle is 5 inches, and the base of the triangle measures 8 inches. Crystal makes a scale drawing of the triangle. If the height of her scale drawing is 2 inches, what is the area of her scale drawing?

(1 point)
Responses

6.4 square inches
6.4 square inches

20 square inches
20 square inches

3.2 square inches
3.2 square inches

1.6 square inches

1 answer

To find the area of Crystal's scale drawing of the triangle, we can first calculate the area of the original triangle and then determine the corresponding area of the scaled version.

The area \( A \) of a triangle is given by the formula:

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

For the original triangle, the base is 8 inches and the height is 5 inches, so:

\[ A = \frac{1}{2} \times 8 \times 5 = \frac{1}{2} \times 40 = 20 \text{ square inches} \]

Now, since Crystal makes a scale drawing where the height is reduced to 2 inches, we can find the scale factor. The scale factor for the height can be calculated as follows:

\[ \text{Scale factor} = \frac{\text{height of scale drawing}}{\text{original height}} = \frac{2}{5} \]

The area of similar figures (such as triangles) scales by the square of the scale factor. Therefore, the area of the scale drawing can be calculated by multiplying the original area by the square of the scale factor:

\[ \text{Area of scale drawing} = A \times \left(\text{Scale factor}\right)^2 = 20 \times \left(\frac{2}{5}\right)^2 \]

Calculating that:

\[ \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]

Now substituting this back into the area calculation:

\[ \text{Area of scale drawing} = 20 \times \frac{4}{25} \]

Calculating the above:

\[ \text{Area of scale drawing} = \frac{80}{25} = 3.2 \text{ square inches} \]

Thus, the area of Crystal's scale drawing is 3.2 square inches.