[ Heron's Formula  [ Area of a Triangle (through semiperimeter and radius of inscribed or exscribed circle).

In a triangle, the sides are in the ratio 2:3:4. A semicircle is inscribed in it with the diameter lying on the largest side. Find the ratio of the area of the semicircle to the area of the triangle.

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If the sides of the triangle are $2 x, 3 x$, and $4 x$, and $r$ is the radius of the given semicircle, then the area of the triangle is $\frac{1}{2}(2 x+3 x) r$.

## Solution

Let the sides of the triangle be $2 x, 3 x$, and $4 x$, and the area be $S$. By Heron's formula,

$$
S=\sqrt{\frac{9 x}{2} \cdot \frac{5 x}{2} \cdot \frac{3 x}{2} \cdot \frac{x}{2}}=\frac{3 x^{2} \sqrt{15}}{4}
$$

On the other hand, if $r$ is the radius of the given circle, then

$$
S=\frac{1}{2}(2 x+3 x) r=\frac{5 x r}{2}
$$

From this, we find that $r=\frac{3 x \sqrt{15}}{10}$. Therefore, if $S_{1}$ is the area of the semicircle, then

$$
S_{1}=\frac{\pi r^{2}}{2}=\frac{27 \pi x^{2}}{40}
$$

Thus,

$$
\frac{S_{1}}{S}=\frac{9 \pi}{10 \sqrt{15}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_8027e864227276d12944g-05.jpg?height=337&width=855&top_left_y=1901&top_left_x=607)

## Answer

$\frac{9 \pi}{10 \sqrt{15}}$