Let G be the center of the equilateral triangle XYZ. A dilation centered at G with scale factor -3/4 is applied to triangle XYZ, to obtain triangle X'Y'Z'. Let A be the area of the region that is contained in both triangles XYZ and X'Y'Z'. Find A/the area of XYZ.
Let the side length of equilateral triangle XYZ be s. Then, the side length of equilateral triangle X'Y'Z' is 3/4s.
The area of triangle XYZ is s^2*sqrt(3)/4.
The area of triangle X'Y'Z' is (3/4)^2*s^2*sqrt(3)/4=9/16s^2*sqrt(3).
The area of the region that is contained in both triangles XYZ and X'Y'Z' is s^2*sqrt(3)/4−9/16s^2*sqrt(3)=5/16s^2*sqrt(3).
Therefore, A/area of XYZ=5/16*s^2*sqrt(3)/s*2*sqrt(3)/4 = 5/16.
