Let DEF be an equilateral triangle with side length 3. At random, a point G is chosen inside the triangle. Compute the probability that the length DG is less than or equal to 1.

Guest Nov 15, 2022

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*Let DEF be an equilateral triangle with side length 3. At random, a point G is chosen inside the triangle. Compute the probability that the length DG is less than or equal to 1.*

Draw equilateral triangle DEF with side length 3.

Place the point of a compass on D and draw a circle with radius 1.

Any point inside the piece of the triangle that's set off by the circle's arc will be 1 or less from D.

Your probability is the area of the piece set off by the arc divided by the area of the entire triangle.

Figure the area of the circle, π r^{2}

then take a sixth of it for the area of the segment. (60/360) • 3**.**14 • 1^{2}

Figure the area of the triangle, (1/2) bh.

You can get the height with Pythagoras' Theorem. h = sqrt(3^{2} – 1**.**5^{2})

My answer: (0**.**524 / 3**.**897) = 0**.**135 or as a percentage **13.5% probability**

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Guest Nov 15, 2022