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# CPhill help pls

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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 $12$?

Jul 29, 2022

#4
+2667
+1

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 $${20 \over 36} = \color{brown}\boxed{5 \over 9}$$

Jul 29, 2022

#1
-1

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.

Jul 29, 2022
#2
-2

It's wrong :(

Guest Jul 29, 2022
#6
+1162
+3

that was a copied answer from wiseowl

nerdiest  Jul 29, 2022
#3
+129390
+3

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  !!!!

Jul 29, 2022
edited by CPhill  Jul 30, 2022
edited by CPhill  Jul 30, 2022
#4
+2667
+1

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 $${20 \over 36} = \color{brown}\boxed{5 \over 9}$$

BuilderBoi  Jul 29, 2022
#5
+1162
+3

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.

nerdiest  Jul 29, 2022
edited by nerdiest  Jul 29, 2022
#7
+2667
+1

Right, but the area of the bottom 2 units is NOT 1/3 of the total area of the triangle.

BuilderBoi  Jul 29, 2022
#8
+1162
+3

hmmmmm

nerdiest  Jul 29, 2022