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Triangle ABC is equilateral with side length 3. A point X is randomly chosen within triangle ABC. What is the probability that X is no more than 2 units away from vertex A?

Dec 7, 2020

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Triangle ABC is equilateral with side length 3. A point X is randomly chosen within triangle ABC. What is the probability that X is no more than 2 units away from vertex A?

Hello Guest!

$$\large P=\frac{A_{sec}}{A_{tri}-A_{sec}}$$

$$A_{sec}= \frac{2^2\pi }{6}\\ A_{tri}= \frac{3^2\sqrt{3}}{4}$$

$$P=\large \frac{ \frac{2^2\pi }{6}}{ \frac{3^2\sqrt{3}}{4} - \frac{2^2\pi }{6}} =\frac{\frac{8\pi }{12}}{\frac{9\cdot 3\sqrt{3}}{12}-\frac{4\cdot 2\pi}{12}} =\frac{8\pi}{27\cdot \sqrt{3}-8\pi}$$

$$P= \large 1.162 :1$$

!

Dec 8, 2020
edited by asinus  Dec 8, 2020