Let $DEF$ be an equilateral triangle with side length $3.$ At random, a point $G$ is chosen inside the triangle. Compute the pro

Question

vesna_86

22 days ago

6

Let $DEF$ be an equilateral triangle with side length $3.$ At random, a point $G$ is chosen inside the triangle. Compute the pro

bability that the length $DG$ is less than or equal to $1.$

Mathematics

2 answers:

AnnZ [9K]22 days ago

8 0

$|\Omega|=(\text{the area of the triangle})=\dfrac{a^2\sqrt3}{4}=\dfrac{3^2\sqrt3}{4}=\dfrac{9\sqrt3}{4}\\|A|=(\text{the area of the sector})=\dfrac{\alpha\pi r^2}{360}=\dfrac{60\pi \cdot 1^2}{360}=\dfrac{\pi}{6}\\\\\\P(A)=\dfrac{\dfrac{\pi}{6}}{\dfrac{9\sqrt3}{4}}\\\\P(A)=\dfrac{\pi}{6}\cdot\dfrac{4}{9\sqrt3}\\\\P(A)=\dfrac{2\pi}{27\sqrt3}\\\\P(A)=\dfrac{2\pi\sqrt3}{27\cdot3}\\\\P(A)=\dfrac{2\pi\sqrt3}{81}\approx13.4\%$

Zina [9.1K] · Answer 1 · 2026-03-06T03:30:47+03:00

Answer:

13.44%

Step-by-step explanation:

For the length of DG to be at most 1, point G needs to be located within a circular sector centered on D, having a radius of 1, and subtending an angle of 60°.

The area of the sector is

$A_s = \dfrac{n}{360^\circ}\pi r^2$