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A triangle is bisected by a vertical line that extends from its vertex to its base. The number 6 is marked near the center of the vertical line. The number 4 is marked near where the vertical line intersects the base of the triangle.

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 and dilated triangles, we can follow these steps:

  1. Calculate the area of the original triangle: 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 4 inches and the height is 6 inches. Therefore: \[ A_{original} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]

  2. Determine the dimensions of the dilated triangle: When Marisol dilates her triangle by a scale factor of 1.5, both the base and height will be scaled by this factor. Thus:

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

  3. Calculate the area of the dilated triangle: Using the same area formula: \[ A_{dilated} = \frac{1}{2} \times \text{new base} \times \text{new height} \] We have: \[ A_{dilated} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]

  4. Compare the areas: Now, we compare the area of the dilated triangle to the original triangle: \[ \frac{A_{dilated}}{A_{original}} = \frac{27}{12} = 2.25 \]

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

In conclusion, the final answer is: The area of the dilated triangle is 2.25 times the area of the original triangle.