Right triangle $XYZ$ has legs of length $XY = 12$ and $YZ = 6$. Point $D$ is chosen at random within the triangle $XYZ$. What is the probability that the area of triangle $XYD$ is at most $12$?
+2668 Wouldn't the probability be the \({\text{successful region} \over \text{total region}} = {[XYAB] \over \triangle XYZ}\)
The area of the successful region is \((2 \times 8) + (4 \times 2 \div 2) = 20\), and the area of the total region is \(12 \times 6 \div 2 = 36\)
So the probability is \({20 \over 36} = \color{brown}\boxed{5 \over 9}\)
In order for the area of triangle XYD to be at most 12, the height of triangle XYD is at most 2. The height of triangle XYD can be 6, so the probability is 2/6 = 1/3.
+129898 XY = the base of the triangle = 12
Then the altitude cannot be more than 2
After some reflection, I believe that BuilderBoi's answer is correct !!!
D can fall ANYWHERE in the area of the trapezoidal area given by
(1/2) (2) (8 + 12) = 20
So.....the probability is 20 / 36 = 5/9
THX BuilderBoi for helping me to see this !!!!
+2668 Wouldn't the probability be the \({\text{successful region} \over \text{total region}} = {[XYAB] \over \triangle XYZ}\)
The area of the successful region is \((2 \times 8) + (4 \times 2 \div 2) = 20\), and the area of the total region is \(12 \times 6 \div 2 = 36\)
So the probability is \({20 \over 36} = \color{brown}\boxed{5 \over 9}\)
+1164 a wise owl told me once
In order for the area of triangle XYD to be at most 12, the height of triangle XYD is at most 2. The height of triangle XYD can be 6, so the probability is 2/6 = 1/3.
WISEOWL's answer, which was voted as a 'best answer'
+2668 Right, but the area of the bottom 2 units is NOT 1/3 of the total area of the triangle.
