00:01
We have let p be a point within the triangle abc, let q be a point on the bottom line of bc of the triangle.
00:06
What is the probability that the line pq and the line ab intersect? so let's say q is like here.
00:26
What is the probability that the line pq and the line ab intersect? and this is just a random example that you can draw.
00:39
Okay, so first of all, for pq and ab to be able to intersect, then q has to be between the projections of a and p onto line bc.
00:56
So we're just gonna call the projections of a and p onto bc as a ' and p'.
01:04
So q has to be somewhere between a ' and p'.
01:12
And since q can be anywhere on bc, the probability of this happening is the ratio of the length of the segment a ' p ' to the length of the segment bc.
01:24
So let's make the whole entire area of the triangle abc.
01:31
Using the formula for the area of triangle, we have abc, which i'm just calling the area, equals 0 .5 1 half base times height.
01:42
So times base, which is bc, times height, where h is the height of the triangle.
01:51
Okay, so now let's consider the area of the triangle ap ' c.
01:59
We'd have 1 half, 0 .5, times the base, which this time is gonna be a ' p ' times the height.
02:12
So if we take the area of the triangle ap ' c and we divide it by the area of the triangle abc, we're gonna end up getting most of it to cancel out.
02:46
So just a ' p ' over bc...