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# geometry

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What is the radius of the circle inscribed in triangle ABC if AB = 12, AC=14, BC=18? Express your answer in simplest radical form.

Sep 5, 2021

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What is the radius of the circle inscribed in triangle ABC if AB = 12, AC=14, BC=18? Express your answer in simplest radical form.

Hello Guest!

The law of cosines:

$$18^2=14^2+12^2-2\cdot 14\cdot 12\cdot cos\ \alpha\\ cos\ \alpha = \frac{14^2+12^2-18^2}{2\cdot 14\cdot 12}=0.\overline{047619}$$

$$\alpha=87.271°\\$$

$$cos\ \beta =\frac{18^2+12^2-14^2}{2\cdot 12\cdot 18}=0.\overline{629}$$

$$\beta=50.977°$$

The functions of the bisector in A and B:

$$f_A(x)=tan\ (\frac{\alpha}{2})\cdot x=0.95346\ x$$

$$f_B(x)=-tan\ (\frac{\beta}{2})\cdot x+12\ tan\ (\frac{\beta}{2}) \\ f_B(x)=-tan\ (25.4886°)\cdot x+12\ tan\ (25.4886°) \\ \color{blue}f_B(x)=-0.47673\ x+5.72078$$

Equate the functions

$$0.95346\ x=-0.47673\ x+5.72078\\ 1.43019\ x=5.72078\\ x=4\\ \color{blue}y=3.81385$$

The radius of the circle inscribed in triangle ABC is 3.81385. !

Sep 6, 2021
edited by asinus  Sep 6, 2021
edited by asinus  Sep 6, 2021