The Pythagorean Theorem

Lesson

Usually we use the Pythagorean Theorem to find the sides lengths of a triangle that we already know is a right triangle. That is, we know that if a given right triangle has shorter side lengths of $a$`a` and $b$`b`, along with a hypotenuse of length $c$`c`, then

$c^2=a^2+b^2$`c`2=`a`2+`b`2

But, we can also use the converse* *of the Pythagorean Theorem to find out for ourselves whether a particular triangle is a right triangle or not. The converse of the theorem says that if the side lengths of a triangle satisfy the above equation, then the triangle must have a right angle.

Converse of the Pythagorean Theorem

If the sum of the squares of the shortest sides of a triangle are equal to the square of the longest side of the triangle, then it is a right triangle.

That is, if a triangle has a longest side with length $c$`c` and the other two sides have lengths $a$`a` and $b$`b` and if

$c^2=a^2+b^2$`c`2=`a`2+`b`2

then the triangle is a right triangle. Otherwise, the triangle is not a right triangle.

When the converse of the Pythagorean Theorem tells us that a triangle is **not **a right triangle, that triangle must either have:

- one obtuse angle (which means it is an obtuse triangle), or
- all acute angles (which means it is an acute triangle).

Just like with right and non-right triangles, we want to be able to tell these two types of triangles apart just by looking at the side lengths of a triangle.

Again we can do this by thinking about the longest side in a triangle. First consider what happens when you change the position of any two sides of a triangle without changing their lengths. What happens to the length of the remaining side? What happens to the angle opposite the remaining side (which is between the two fixed ones)? For example, think about what happens to the triangle formed by the two hands of a clock as the hands change their position.

As the two sides move closer together, the remaining side length will get smaller and so will the size of the angle opposite it. This reflects the fact that an angle of a triangle is related to the size of the side length opposite it.

In particular, the longest side length will always be opposite the largest angle in the triangle. So, if you increase the length of a triangle's longest size, the largest angle will be bigger. Likewise, if the longest side gets shorter then the largest angle will be smaller.

So if the longest side is long enough, then the largest angle will be obtuse (and so will the triangle), If the longest side is short enough, then the largest angle will be acute (and so will the triangle). If the longest side is *just* the right size (that is, it satisfies the Pythagorean Theorem), then the largest angle with be a right angle and the triangle will be a right triangle.

We can summarize all of these facts, therefore, by comparing the size of the longest side in a triangle in terms of the Pythagorean Theorem.

Pythagorean Results

If a triangle has a longest side with length $c$`c` and the other two sides have lengths $a$`a` and $b$`b`, then we have the following results:

The advantage of the above results is that you don't actually need to see a triangle to know what it looks like. You can get all of this information just from the numbers that indicate side lengths. However this can also be a problem, since it might be the case that a certain set of side lengths can't actually form any triangle at all.

To check whether three lengths can form a triangle at all, we first need to check that they satisfy the triangle inequality.

The Triangle Inequality

For any triangle, the sum of any two side lengths must be greater than the remaining side length.

That is, if $a$`a`, $b$`b` and $c$`c` are all side lengths in a triangle, then $a+b>c$`a`+`b`>`c`.

Consider a triangle whose shortest sides have lengths $6$6 and $8$8. The longest side of the triangle has a length of $c$`c`.

What must the value of $c$

`c`be if the triangle has a right angle?If $c=12$

`c`=12, then which of the following correctly describes the triangle?Right angled

AObtuse

BAcute

CRight angled

AObtuse

BAcute

CIf $c=9$

`c`=9, then which of the following correctly describes the triangle?Acute

AObtuse

BRight angled

CAcute

AObtuse

BRight angled

C

Consider three straight lines with lengths $9$9, $12$12 and $14$14 units.

Is it possible to form a triangle using these lines?

Yes

ANo

BYes

ANo

BWhich of the following correctly describes the triangle formed?

Obtuse triangle

AAcute triangle

BRight-angled triangle

CObtuse triangle

AAcute triangle

BRight-angled triangle

CWhich of the following inequalities justifies your answer in part (b)?

$9^2+12^2>14^2$92+122>142

A$9+12<14$9+12<14

B$9+12>14$9+12>14

C$9^2+12^2<14^2$92+122<142

D$9^2+12^2>14^2$92+122>142

A$9+12<14$9+12<14

B$9+12>14$9+12>14

C$9^2+12^2<14^2$92+122<142

D

Solve problems using the Pythagorean Theorem, as required in applications