If the right triangular prism is dilated using a scale factor of 3, what would be the volume of the dilated prism? (1 point)

sides are 3 cm, 4 cm, 7 cm

1 answer

To find the volume of the dilated right triangular prism, we first need to determine the volume of the original prism and then apply the dilation.

  1. Finding the area of the triangular base: The triangle formed by the sides 3 cm, 4 cm, and the height corresponding to the base of 4 cm. We can use the formula for the area of a right triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, we can consider the base as 4 cm and the height as 3 cm: \[ \text{Area} = \frac{1}{2} \times 4 \times 3 = \frac{12}{2} = 6 \text{ cm}^2 \]

  2. Finding the volume of the original prism: The volume \( V \) of a prism is given by the formula: \[ V = \text{Base Area} \times \text{Height} \] In this case, the height of the prism is 7 cm: \[ V = 6 \times 7 = 42 \text{ cm}^3 \]

  3. Applying the scale factor: When a prism is dilated with a scale factor of \( k \), its volume changes by a factor of \( k^3 \). In this case, \( k = 3 \): \[ \text{New Volume} = \text{Original Volume} \times k^3 = 42 \times 3^3 = 42 \times 27 = 1134 \text{ cm}^3 \]

Thus, the volume of the dilated triangular prism would be 1134 cm³.