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# Geometry

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Triangle ABC has side lengths 29, 40, and 40.  Find the inradius of triangle ABC.  Find the circumradius of triangle ABC.

Nov 6, 2021

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Triangle ABC has side lengths c = 29, a = 40, and b = 40. Find the inradius of triangle ABC. Find the circumradius of triangle ABC.

Hello Guest!

$$sin( \dfrac{\gamma}{2} )=\dfrac{c}{2a}=\dfrac{29}{2\cdot40}\\ \dfrac{\gamma}{2}=21,2538°\\ \color{blue}\gamma =42.5076°\\ \color{blue}\alpha =\beta =68.4762°$$

$$f(x)=tan(\frac{\alpha}{2})x=tan(34.2381°)\cdot x\\ x=\dfrac{c}{2}=14.5\\ r_i=tan(34.2381°)\cdot 14.5\\ \color{blue}r_i=9.918\\$$

$$m\ of\ \overline{AC}=tan(68.4762)\\ m\ of\ g(x)= -\dfrac{1}{tan(68.4762)}\\ g(x)=-\dfrac{1}{tan(68.4762)}\cdot (x-29)\\ x=14.5\\ g(x)=5.640$$

$$h=\sqrt{a^2-\dfrac{c^2}{4}}=\sqrt{40^2-\frac{29^2}{4}}\\ \color{blue}h=37.279$$

$$r_c=h-g(x)=37.279-5.640\\ \color{blue}r_c=31.639$$

The inradius of triangle ABC is 9.918.

The circumradius of triangle ABC is 31.639.

!

Nov 7, 2021