A rectangle, whose one sidelength is twice the other side, is inscribed inside a triangles with sides $3$ cm, $4$ cm and $5$ cm, such that the long sides lies entirely on the long side of the triangle. The other two remaining vertices of the rectangle lie respectively on the other two sides of the triangle. Find the lengths of the sides of this rectangle.
Problem
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Tags: geometry, rectangle, right triangle, inscribed
i3435
23.03.2020 21:34
Use similar triangles.
os31415
23.03.2020 22:37
Is this a computational contest?
name the short side of the rectangle $5x$, and the long side $10x$. The small triangle sharing a right angle with the 3-4-5 triangle is similar to it because of parallel lines. Therefore, it is a $6x-8x-10x$ triangle. The triangle on the upper left with its hypotenuse on the leg with length 3 is similar because of AA similarity. By leg ratios, $\frac{5x}{4}=\frac{3-6x}{5}\implies{49x=12}\implies{x=\frac{12}{49}}\implies{5x=\frac{60}{49}}\implies{10x=\frac{120}{49}}$. The lengths of the sides are $\frac{60}{49}, \frac{120}{49}$
420th post
parmenides51
23.03.2020 22:49
It comes from an Open Contest for Juniors in Estonia, source
skrublord420
24.03.2020 06:13
os31415 wrote: Is this a computational contest?
name the short side of the rectangle $5x$, and the long side $10x$. The small triangle sharing a right angle with the 3-4-5 triangle is similar to it because of parallel lines. Therefore, it is a $6x-8x-10x$ triangle. The triangle on the upper left with its hypotenuse on the leg with length 3 is similar because of AA similarity. By leg ratios, $\frac{5x}{4}=\frac{3-6x}{5}\implies{49x=12}\implies{x=\frac{12}{49}}\implies{5x=\frac{60}{49}}\implies{10x=\frac{120}{49}}$. The lengths of the sides are $\frac{60}{49}, \frac{120}{49}$
420th post Congrats on your 420th post. You are now the top DOGG.