Triangles (1 Viewer)

Mongoose528 · Jul 29, 2016

Of a right-angled triangle with side lengths of 6 and 8 cm, what is the raidus of the largest circle that can fit inside of it?

thanks in advance

InteGrand · Jul 29, 2016

Mongose528 said:
Of a right-angled triangle with side lengths of 6 and 8 cm, what is the raidus of the largest circle that can fit inside of it?

thanks in advance

$\noindent The answer is the inradius (radius of the incircle). Let the inradius be $r$, the semiperimeter of the triangle be $s$, and the area of the triangle be $A$. Then $A = rs$ (this holds for a general triangle; you can show this result by drawing the perpendicular a from the incentre (centre of the incircle) to the three sides and connecting each of the triangle's vertices to the incentre. The area of each of these three smaller triangles formed is $\frac{1}{2}r x$, where $x$ is the base length of that triangle. Add these up and you'll get the result).$

$\noindent For this triangle, the area $A$ is $\frac{1}{2}\times 6\times 8 = 24$ and the semiperimeter is $\frac{1}{2}\times (6+8+10) = 12$ (as the third side is $10$ by Pythagoras's theorem). So $r =\frac{A}{s} = \frac{24}{12} = 2$. So answer is $2$ cm.$

You can find more info on incircles here: http://mathworld.wolfram.com/Incircle.html .

Mongoose528 · Jul 29, 2016

Thanks

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Triangles (1 Viewer)

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