Question

A triangle with vertices at A(0, 0), B(0, 4), and C(16, 0) is dilated to yield a triangle with vertices at A'(0, 0), B(0, 10), and C(15, 0). The origin is the center of dilation. What is the scale factor of the dilation?

a) 2
b) 1.5
c) 0.5
d) 0.75

Answer 1

The stated dilation scale factors for points B and C in the triangle are inconsistent, with a vertical scale factor of 2.5 and a horizontal scale factor of approximately 0.9375. There might be a typo in the question as a valid dilation should have a consistent scale factor across all coordinates.

Step-by-step explanation:

To determine the scale factor of the dilation, we can compare the corresponding sides of the original triangle and the dilated triangle. Initially, point B is at (0, 4) and moves to (0, 10) after dilation, while point C is initially at (16, 0) and moves to (15, 0). Here, we look at the y-coordinate change for point B because the x-coordinate remains at 0, which means the triangle is only vertically stretched, not horizontally. The original distance from A to B in the y-direction is 4 units and after dilation, it is 10 units.

To find the scale factor, we divide the new distance by the original distance:
Scale Factor = New distance / Original distance = 10 / 4 = 2.5.

However, this contradicts with the information given for point C, which after dilation became (15, 0) - which would imply a horizontal scale factor of 15 / 16 (which is not equal to 2.5). Hence, there seems to be a discrepancy in the dilation of points B and C, suggesting that maybe there is a typo regarding the coordinates of C after dilation. If we assume correct dilation, the scale factor would be consistent across all points of the triangle. Examining the scale factor for point C's x-coordinate transformation, we have:

Horizontal Scale Factor = New x-coordinate of C / Original x-coordinate of C = 15 / 16 = 0.9375.

Therefore, the question contains an inconsistency in the dilation of point C, or there is a typo in the coordinates for point C.