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Let G be the center of equilateral triangle XYZ  A dilation centered at G with scale factor-7/5  is applied to triangle XYZ to obtain triangle X'Y'Z' Let A be the area of the region that is contained inside both triangles XYZ and X'Y'Z' Find A/[XYZ]

 May 24, 2024
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Consider the equilateral triangle XYZ. Let the side length of this triangle be s Then the area of the triangle is

 

K=3​s2/4

 

The dilation with center G and scale factor −57​ reduces side lengths by a factor of ​−57​​=57​ The area of the scaled triangle, X′Y′Z′, is then K′=(57​)2K=2549​K

 

The overlapping region is the original triangle minus the scaled triangle. Since the scaling reduces all side lengths equally, the resulting shrunken triangle is similar to the original triangle.

 

Therefore, the ratio of the area of the shrunken triangle to the original triangle is the square of the ratio of the side lengths. This ratio is (57​)2=2549​ , so the area of the overlapping region is K−2549​K = 6/25*K

 

The ratio of the area of the overlap to the original triangle is then A/K = 6/25.

 May 25, 2024

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