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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.

 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 r2  

then take a sixth of it for the area of the segment.    (60/360) • 3.14 • 12   

 

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

You can get the height with Pythagoras' Theorem.    h = sqrt(32 – 1.52)  

 

My answer:   (0.524 / 3.897) = 0.135  or as a percentage  13.5% probability   

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 Nov 15, 2022

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