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High School Math Grades 9-12, Ages 13-18

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#1 h Sep 14, 2021, 4:29 PM

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In triangle $ABC$ , side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$ ?

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#2 h Sep 14, 2021, 6:36 PM

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Let $A(0,0,B(a,b),C(8,0)$ .

Points on $AB\ :\ D(k,\frac{bk}{a}),F(l,\frac{bl}{a})$ with $k < l$ .

Following the conditions $k=a(1-\frac{\sqrt{6}}{3}),l=a(1-\frac{\sqrt{3}}{3}$ .
The length of the segment $DE$ equals $\frac{8\sqrt{6}}{3}$ , the length of the segment $FG$ equals $\frac{8\sqrt{3}}{3}$ .

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