A straight line parallel to the base of a triangle divides it into parts whose areas are 2 : 1.

Question

A straight line parallel to the base of a triangle divides it into parts whose areas are 2 : 1. In what respect, counting from the top, does it divide the sides?

So what I understand here is that the ratio of the area of two similar triangles are equal to the square of the ratio of their corresponding sides. Applying this to the question will give me $\sqrt2 : 1$, which is not same as the answer given.

Answer given : $(\sqrt6 +2) : 1 $ or $(\sqrt3 +1) : 2$.

Somebody help me here. Thanks.

Answer 1

If the trapezium is larger than the small triangle, then The ratio of the areas of the similar triangles is 3:1

The ratio of the lengths of the triangles are rt3:1

The ratio of the sides are then rt 3 -1 : 1. Multiplying both sides by rt 3 +1, we get 2:rt3 +1.

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Now do the same if the trapezium has area smaller than the inner triangle.

Answer 2

$$ \frac{x^2}{1-x^2}=2\implies x=\sqrt\frac23\implies \frac x{1-x}=\frac{\sqrt\frac23}{1-\sqrt\frac23}=\frac{\sqrt\frac23\left(1+\sqrt\frac23\right)}{\frac13}=\sqrt6+2. $$ and similarly for the case $\frac{x^2}{1-x^2}=\frac12.$