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The area of a right triangle is equal to $2 \sqrt{3}$. Determine its height projected to the hypotenuse if it divides the right angle in a ratio of 1:2.

I don't really understand how to obtain the height projected to the hypotenuse here. I've tried using the formulas $S = \frac12ab$ and $S= \frac12ch_c$ where a and b are the legs, c is the hypotenuse and $h_c$ is the height projected to the hypotenuse. Answer given is $\sqrt3$.

Can anybody help me here?

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2 Answers 2

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The altitude to the hypotenuse of a right triangle divides the triangle into two similar copies of itself. So knowing that the altitude divides the right angle in the ratio $1:2$, we conclude that the right triangle's acute angles are $30^\circ$ and $60^\circ$.

Let $x$ and $x\sqrt3$ represent the lengths of the legs of the right triangle. We know that the area of the triangle is $\frac12\cdot x\cdot x\sqrt 3=2\sqrt3$, so $\frac12 x^2=2$ and thus $x=2$. The hypotenuse of the triangle is therefore $4$ and the height to the hypotenuse is given by $\frac12\cdot 4\cdot h=2\sqrt 3$ or $h=\sqrt3$.

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  • $\begingroup$ Ok. I understand the concept here. But why did you let x and x sqrt(3) to represent the legs of the triangle? $\endgroup$
    – ThermalRaindrops62
    Commented Nov 18, 2019 at 13:53
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    $\begingroup$ @ThermalRaindrops62 Because it is a $30^\circ-60^\circ-90^\circ$ right triangle, we know that the sides are in the proportion $1:\sqrt3:2$. $\endgroup$
    – user694818
    Commented Nov 18, 2019 at 20:10
  • $\begingroup$ Oh ok got it now. Tq. $\endgroup$
    – ThermalRaindrops62
    Commented Nov 18, 2019 at 23:20
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enter image description here

$$S = h \cdot (a + b) / 2$$ $$h^2 = a \cdot b$$ $$h = a\sqrt{3}$$

Then

$$b = 3a$$ $$S = 2a^2\sqrt{3} = 2\sqrt{3}$$ $$a = 1$$ $$h = \sqrt{3}$$

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  • $\begingroup$ hmm.. why did you let b = 3a here when the ratio is 1:2? $\endgroup$
    – ThermalRaindrops62
    Commented Nov 18, 2019 at 14:27
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    $\begingroup$ all triangles are similar and in all angles are 30, 60 and 90 degrees, so legs are $\sqrt{3} : 1$, therefore since $\frac{h}{a} = \frac{b}{h} = \sqrt{3}$, then b = 3a $\endgroup$
    – Anatoliy R
    Commented Nov 18, 2019 at 14:32
  • $\begingroup$ Sorry, but I still don't understand how you get √3 $\endgroup$
    – ThermalRaindrops62
    Commented Nov 18, 2019 at 15:18
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    $\begingroup$ because right angle is divided with ratio 1:2, so angles are 30 and 60 degrees. In such triangles hypotenuse ratio to smaller leg is 2:1, so the other leg is $\sqrt{3}$ $\endgroup$
    – Anatoliy R
    Commented Nov 18, 2019 at 15:35

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