Question

ΔABC is dilated by a scale factor of 0.5 with the origin as the center of dilation, resulting in the image ΔA′B′C′. If A = (2, 2), B = (4, 3), and C = (6, 3), what is the length of the triangle ? A. 3 units B. 4 units C. 2 units

D. 1 unit

Answer 1

After dilating ∆ABC by a scale factor of 0.5, the length of side AB' is approximately 1.118 units. This means the closest whole unit for the length of the triangle, as measured from AB', is 1 unit.

Step-by-step explanation:

You have been asked to find the new coordinates of the vertices of ∆ABC after a dilation with a scale factor of 0.5 and to determine the length of the triangle after the dilation. Since the dilation is with the origin as the center, you simply multiply each coordinate of the vertices by the scale factor 0.5. Here are the calculations:

• A' = (2 × 0.5, 2 × 0.5) = (1, 1)
• B' = (4 × 0.5, 3 × 0.5) = (2, 1.5)
• C' = (6 × 0.5, 3 × 0.5) = (3, 1.5)

To find the length of the triangle, which can be interpreted as the length of one of the sides, let's take AB'. The original length of AB without dilation is found using the distance formula: √((4-2)² + (3-2)²) = √(4 + 1) = √5 ≈ 2.236. After dilation, this length is halved: √5 / 2 ≈ 1.118. However, since the answer must be in whole units, the closest answer in units is 1 unit.