A triangle has sides with lengths of 4, 9, and 8. What is the radius of the triangles inscribed circle?

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A triangle has sides with lengths of 4, 9, and 8. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-02-09T20:54:00

$1.524$ $1.524$

Explanation:

Refer to the figure below

I created this figure using MS Excel

As the sides of the triangle are 4, 8 and 9:
$x + y = 9$ $x+y=9$
$x + z = 8$ $x+z=8$
$y + z = 4$ $y+z=4$ => $z = 4 - y$ $z=4-y$
$\to x + 4 - y = 8$ $-> x+4-y=8$ => $x - y = 4$ $x-y=4$

Adding the first and last equations
$2 x = 13$ $2x=13$ => $x = 6.5$ $x=6.5$

Using the Law of Cosines ( $△_{A B C}$ $triangle_(ABC)$ ):
$4^{2} = 9^{2} + 8^{2} - 2 \cdot 9 \cdot 8 \cdot cos α$ $4^2=9^2+8^2-2*9*8*cos alpha$

$cos α = \frac{81 + 64 - 16}{144} = \frac{129}{144}$ $cos alpha=(81+64-16)/144=129/144$

$α = {26.384}^{\circ}$ $alpha=26.384^@$

In the right triangle with $x$ $x$ as cathetus, we can see that
$tan (\frac{α}{2}) = \frac{r}{x}$ $tan (alpha/2)=r/x$

$r = 6.5 \cdot tan (\frac{{26.384}^{\circ}}{2})$ $r=6.5*tan (26.384^@/2)$ => $r = 1.524$ $r=1.524$

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A triangle has sides with lengths of 4, 9, and 8. What is the radius of the triangles inscribed circle?

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Explanation:

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