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Given: X, Y, and Z are the midpoints of the sides of \triangle ABC. Find the ratio of the area...

Question:

Given: X, Y, and Z are the midpoints of the sides of {eq}\triangle ABC. {/eq} Find the ratio of the area of {eq}\triangle XYZ \text{ to the area of } \triangle ABC. {/eq}

Similar Triangles.

Triangles are similar if they have the same shape and proportions, but can be different in size. When one of the triangles is enlarged to produce the second triangle, each part of the original will correspond to a particular part of the new triangle. In two similar triangles, corresponding angles are equal and corresponding sides have the same ratio. One of the criteria of similarity is the SSS criteria : If the corresponding sides of two triangles are proportional, then the two triangles are similar. In similar triangles where the ratio of corresponding sides is {eq}k {/eq}, the ratio of areas is {eq}k^2 {/eq}.

Answer and Explanation: 1

For the midpoint theorem we have that {eq}\displaystyle XY=\frac{1}{2} AC {/eq}, {eq}\displaystyle YZ=\frac{1}{2} AB {/eq} and {eq}\displaystyle XZ=\frac{1}{2} BC {/eq}. Therefore the two triangles {eq}XYZ {/eq} and {eq}ABC {/eq} have proportional sides; for the SSS criteria they are similar, with scale factor equal to {eq}\displaystyle \frac{1}{2}{/eq}.

The ratio of the area of {eq}\triangle XYZ \text{ to the area of } \triangle ABC. {/eq} is {eq}\displaystyle \frac{1}{4} {/eq}.


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