Question

Given the circle with radius 12, what is the probability of choosing a point inside the triangle?

Answer 1

This one looks fun. To find the probability we need
P = area of triangle/ area of circle

Answer 2

Let's start with the circle. Do you know the area formula for any circle?

Answer 3

no

Answer 4

A = pi * r^2

Answer 5

ok

Answer 6

so what area do you get?

Answer 7

113.09

Answer 8

not quite.
A = 3.14*12*12= ?

Answer 9

452.16

Answer 10

sooo?

Answer 11

there we go. now we have the tricky part of the triangle. Since I assume that is a central angle we know it is the same angle in the triangle as the arc. See attached picture.

Answer 12

ok

Answer 13

Our triangle area is
\[A = 1/2 b*h\]
we have to solve for both of those first and we can use the 30 - 60- 90 pattern

Answer 14

ok

Answer 15

that gives us h= 6
and half the base is 6sqrt3 (since we split the triangle in half)
The total base is 12sqrt3

Answer 16

ok so how do we turn that into prob.?

Answer 17

We have to finish with the area of the triangle first.. then the probability is
area triangle / area circle
the area of the triangle ends up being 36 * sqrt 3

Answer 18

So the probability is
\[P=36\sqrt{3}/452.16\]

Answer 19

thank you sooo much

Answer 20

anytime