Question

use the image to answer the question.
a right triangle with base and height 3 cm is dilated using a scale factor of 6. what is the ratio of the area of the dilated triangle to the area of the original triangle? use a decimal response if necessary.
the ratio of the areas is:

Explanation:

Step1: Recall area of triangle formula

The area of a right triangle is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} \). For the original triangle, base \( b = 3 \) cm and height \( h = 3 \) cm, so original area \( A_{1}=\frac{1}{2}\times3\times3=\frac{9}{2} \) \( \text{cm}^2 \).

Step2: Find dimensions of dilated triangle

When a figure is dilated by a scale factor \( k \), all linear dimensions (base, height) are multiplied by \( k \). Here, scale factor \( k = 6 \), so the new base \( b_{2}=3\times6 = 18 \) cm and new height \( h_{2}=3\times6 = 18 \) cm.

Step3: Calculate area of dilated triangle

Using the area formula for the dilated triangle, \( A_{2}=\frac{1}{2}\times b_{2}\times h_{2}=\frac{1}{2}\times18\times18 = 162 \) \( \text{cm}^2 \).

Step4: Find the ratio of areas

The ratio of the area of the dilated triangle to the original triangle is \( \frac{A_{2}}{A_{1}}=\frac{162}{\frac{9}{2}} \). When dividing by a fraction, we multiply by its reciprocal: \( \frac{162\times2}{9}= \frac{324}{9}=36 \).

(Alternatively, we can use the property that when a figure is dilated by a scale factor \( k \), the ratio of the areas of the dilated figure to the original figure is \( k^{2} \). Here, \( k = 6 \), so the ratio is \( 6^{2}=36 \), which is a more direct method.)

Answer:

36