+18 Let G be the center of equilateral triangle 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]
+1633 To solve the problem, let's break it down step by step:
Step 1: Understand the Dilation
The dilation is centered at GG, with a scale factor of −75-\frac{7}{5}.
A negative scale factor means the resulting triangle X′Y′Z′X'Y'Z' is flipped across the center GG.
The magnitude of the scale factor (75\frac{7}{5}) indicates the relative size of the dilated triangle compared to the original triangle.
Step 2: Ratio of Areas
The area of a dilated figure changes by the square of the scale factor. Thus:
Area of X′Y′Z′=(75)2⋅Area of XYZ=4925⋅Area of XYZ\text{Area of } X'Y'Z' = \left( \frac{7}{5} \right)^2 \cdot \text{Area of } XYZ = \frac{49}{25} \cdot \text{Area of } XYZ
Step 3: Overlap of Triangles
Since the dilation is centered at GG, the two triangles share the same center and are similar.
The region of overlap forms a smaller triangle that is geometrically similar to both XYZXYZ and X′Y′Z′X'Y'Z',
with sides proportionate to the smaller absolute scale factor 75−1=25\frac{7}{5} - 1 = \frac{2}{5}.
Step 4: Area of Overlap
The overlap triangle has sides scaled by 25\frac{2}{5}, so its area is proportional to the square of this scale:
Area of Overlap=(25)2⋅Area of XYZ=425⋅Area of XYZ\text{Area of Overlap} = \left( \frac{2}{5} \right)^2 \cdot \text{Area of } XYZ = \frac{4}{25} \cdot \text{Area of } XYZ
Step 5: Ratio A[XYZ]\frac{A}{[XYZ]}
The ratio of the area of the overlap to the area of triangle XYZXYZ is:
A[XYZ]=425⋅Area of XYZArea of XYZ=425\frac{A}{[XYZ]} = \frac{\frac{4}{25} \cdot \text{Area of } XYZ}{\text{Area of } XYZ} = \frac{4}{25}
Final Answer:
4/25
