10.01 The Pythagorean theorem and its converse

Lesson

Review the Pythagorean theorem

A long time ago, a Greek mathematician named Pythagoras came up with an interesting observation about right triangles: if $c$c represents the length of the hypotenuse, and $a$a and $b$b represent the other two sides (that meet at a right angle), then $a^2+b^2=c^2$a2+b2=c2.

In other words, the square of the hypotenuse ($c$c) of a right triangle is equal to the square of side $a$a plus the square of side $b$b.

The following interactive demonstrates using areas Pythagoras' Theorem.  You can use the slider and move the squares of a and b to cover the area of c squared, thus demonstrating that $a^2+b^2=c^2$a2+b2=c2

(Watch this video to see the interactive in action -  )

Pythagorean theorem

If $\triangle ABC$ABC is a right triangle, then

$a^2+b^2=c^2$a2+b2=c2,

where $c$c is the hypotenuse of the triangle.

Practice questions

Question 1

Calculate the value of $c$c in the triangle below.

Question 2

Calculate the value of $b$b in the triangle below.

Question 3

$VUTR$VUTR is a rhombus with perimeter $112$112 cm. The length of diagonal $RU$RU is $46$46 cm.

1. First find the length of $VR$VR.

2. Then find the length of $RW$RW.

3. If the length of $VW$VW is $x$x cm, find $x$x correct to 2 decimal places.

4. Hence, what is the length of the other diagonal $VT$VT correct to 2 decimal places.

The converse of the Pythagorean theorem

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

$c^2=a^2+b^2$c2=a2+b2

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 $\triangle ABC$ABC has a longest side with length $c$c and the other two sides have lengths $a$a and $b$b and

$c^2=a^2+b^2$c2=a2+b2

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

Obtuse and Acute Triangles

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.

Investigate: Side lengths and angles

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:

• If $c^2>a^2+b^2$c2>a2+b2 then the triangle is an obtuse triangle.
• If $c^2=a^2+b^2$c2=a2+b2 then the triangle is a right triangle.

D

Outcomes

II.G.SRT.8

Use trigonometric ratios and the pythagorean theorem to solve right triangles in applied problems.