Question

Triangle DEF has vertices D(-4,1), E(-3,5), and F(-2,0) and is dilated by a factor of 1/4 using the point (1,4) as the point of dilation. The dilated triangle is named D' E' F' . What are the coordinates of the vertices of the resulting triangle?

Answer 1

To find the new coordinates of the vertices of the dilated triangle, we can use the formula D' = (1/4)(D - P) + P, where D' is the new coordinate, D is the original coordinate, and P is the center of dilation. Applying this formula to each vertex of the original triangle, we find that the new coordinates are (0, 17/4), (-1/4, 13/4), and (0/4, 5/4).

Step-by-step explanation:

To dilate a triangle by a factor of 1/4 using a point as the center of dilation, we can use the formula:

D' = (1/4)(D - P) + P

where D' is the new coordinate of the point D, D is the original coordinate of the point D, P is the center of dilation, and the operations are scalar multiplication and vector addition.

Using this formula, we can calculate the new coordinates of the vertices:

D' = (1/4)(-4,1 - (1,4)) + (1,4)

E' = (1/4)(-3,5 - (1,4)) + (1,4)

F' = (1/4)(-2,0 - (1,4)) + (1,4)

Calculating the above expressions, we get:

D' = (-4/4, 1/4) + (1,4) = (-1, 1/4) + (1,4) = (0, 17/4)

E' = (-3/4, 5/4) + (1,4) = (-2/4, 9/4) + (1,4) = (-1/4, 13/4)

F' = (-2/4, 0/4) + (1,4) = (-1/4, 4/4) + (1,4) = (0/4, 5/4)