Triangle XYZ is dilated by a factor 1/2. What is the ratio of the area of triangle XYZ to the area of it's image, triangle X'Y'Z'?
2:1
actually i think its 1:2
To find the ratio of the areas of two similar figures, you need to square the ratio of their corresponding side lengths.
In this case, triangle XYZ is dilated by a factor of 1/2. This means that all the side lengths of triangle XYZ are multiplied by 1/2 to get the corresponding side lengths of triangle X'Y'Z'.
Let's assume that the side lengths of triangle XYZ are a, b, and c, and the side lengths of triangle X'Y'Z' are a', b', and c'. The corresponding side lengths are related by the factor of dilation:
a' = (1/2) * a
b' = (1/2) * b
c' = (1/2) * c
To find the ratio of the areas, you need to square the ratio of the corresponding side lengths. In this case, it is (1/2) * a / a = 1/2, (1/2) * b / b = 1/2, and (1/2) * c / c = 1/2.
So, the ratio of the areas of triangle XYZ to triangle X'Y'Z' is (1/2)^2 : 1 = 1/4 : 1 = 1 : 4.
Therefore, the ratio of the area of triangle XYZ to the area of its image, triangle X'Y'Z', is 1 : 4.