Random Triangles

Suppose that a stick is randomly broken in two places. What is the probability that the three pieces form a triangle?

Without looking below, make a guess.

Run the triangle experiment 50 times. Do not be concerned with all of the information displayed in the applet, but just note whether the pieces form a triangle. Would you like to revise you guess in Exercise 1?

Mathematical Formulation

As usual, the first step is to model the random experiment mathematically. We will take the length of the stick as our unit of length, so that we can identify the stick with the interval

01

. To break the stick into three pieces, we just need to select two points in the interval. Thus, let

X

denote the first point chosen and

Y

the second point chosen. Note that

X

and

Y

are random variables and hence the sample space of our experiment is

Now, to model the statement that the points are chosen at random, let us assume, as in the previous sections, that

X

and

Y

are independent and each is uniformly distributed on

01

Show that $X Y$ is uniformly distributed on $S 012$ .

The Probability of a Triangle

Argue that the three pieces form a triangle if and only if the triangle inequalities hold: the sum of the lengths of any two pieces must be greater than the length of the third piece.

Show that the event that the pieces form a triangle is $T T 1 T 2$ where

$T 1 x y S y 1 2 x 1 2 y x 1 2$
$T 2 x y S x 1 2 y 1 2 x y 1 2$

A sketch of the event

T

is given below. Curiously,

T

is composed of triangles!

How close did you come with your guess in Exercise 1? The relative low value of the probability in Exercise 6 is a bit surprising.

Run the triangle experiment 1000 times, updating every 10 runs. Note the apparent convergence of the empirical probability of $T c$ to the true probability.

Triangles of Different Types

Now let us compute the probability that the pieces form a triangle of a given type. Recall that in an acute triangle all three angles are less than 90°, while an obtuse triangle has one angle (and only one) that is greater than 90°. A right triangle, of course, has one 90° angle.

Suppose that a triangle has side lengths $a$ , $b$ , and $c$ , where $c$ is the largest of these. Recall (or show) that the triangle is

acute if and only if $c 2 a 2 b 2$ .
obtuse if and only if $c 2 a 2 b 2$ .
right if and only if $c 2 a 2 b 2$ .

Part (c), of course, is the famous Pythagorean theorem, named for the ancient Greek mathematician Pythagoras.

Show that the right triangle equations for the stick pieces are

$y x 2 x 2 1 y 2$ in $T 1$
$1 x 2 x 2 y x 2$ in $T 1$
$x 2 y x 2 1 y 2$ in $T 1$
$x y 2 y 2 1 x 2$ in $T 2$
$1 x 2 y 2 x y 2$ in $T 2$
$y 2 x y 2 1 x 2$ in $T 2$

Let $R$ denote the event that the pieces form a right triangle. Show that $R 0$

Show that the event that the pieces form an acute triangle is $A A 1 A 2$ where

$A 1$ is the region inside curves (a), (b), and (c) of Exercise 9.
$A 1$ is the region inside curves (d), (e), and (f) of Exercise 9.

Show that the event that the pieces form an obtuse triangle is $B B 1 B 2 B 3 B 4 B 5 B 6$ where

$B 1$ , $B 2$ , and $B 3$ are the regions inside $T 1$ and outside of curves (a), (b), and (c) of Exercise 9, respectively.
$B 4$ , $B 5$ , and $B 6$ are the regions inside $T 2$ and outside of curves (d), (e), and (f) of Exercise 9, respectively.

Show that

$B 1 x 0 1 2 x 1 2 x 2 2 x 3 8 22$
$B 2 x 0 1 2 x 1 2 x 2 2 x 3 8 22$
$B 3 y 0 1 2 y 1 2 y 3 2 3 8 22$

Argue from symmetry that

B 9 4 32 0.1706

You might also argue from symmetry that $B i$ must be the same for each $i$ , even though $B 1$ and $B 2$ (for example) are not congruent.

Show that

A 322 0.07944

Run the triangle experiment 1000 times, updating every 10 runs. Note the apparent convergence of the empirical probabilities to the true probabilities.

3. Random Triangles

Preliminaries

Statement of the Problem

Mathematical Formulation

The Probability of a Triangle

Triangles of Different Types