Marisol draws a triangle with a 4-inch base and 6-inch height. She dilates the triangle by a scale factor of 1.5. Compare the areas of the dilated triangle and the original triangle. Use a decimal response if necessary. (1 point) The area of the dilated triangle is times the area of the original triangle

1 answer

To find the areas of the original triangle and the dilated triangle, we can use the formula for the area of a triangle:

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

Step 1: Calculate the area of the original triangle.

Given:

  • Base = 4 inches
  • Height = 6 inches

\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]

Step 2: Calculate the dimensions of the dilated triangle.

The scale factor is 1.5, so we multiply the base and the height of the original triangle by this factor.

  • Dilated base = \(4 \times 1.5 = 6\) inches
  • Dilated height = \(6 \times 1.5 = 9\) inches

Step 3: Calculate the area of the dilated triangle.

\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]

Step 4: Compare the areas of the dilated triangle and the original triangle.

To find how many times the area of the original triangle fits into the area of the dilated triangle:

\[ \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{27}{12} = 2.25 \]

Thus, the area of the dilated triangle is 2.25 times the area of the original triangle.

The final answer is:

2.25