04Apythagoras

toc = Pythagoras' Theorem =

Pythagoras was a Greek mathematician who started a school (for adults) that studied mathematics, science and philosophy. He was born on the Greek Island of Samos in approximately 570 BC and lived for about 75 years.

The Pythagorean Theorem has been named after Pythagoras since about 50 BC (probably earlier). Indian and Babylonian mathematicians were using the idea well before the time of Pythagoras. While he definitely knew about it and used it, Pythagoras didn't "discover" the theorem.

Hypotenuse
{pronounced: ** hi-POT-en-use ** } In any right angled triangle, the ** hypotenuse ** is
 * the side opposite the right angle
 * the longest side

Pythagoras' Theorem
If we label a right angled triangle so that
 * ** c ** is the length of the hypotenuse
 * ** a ** and ** b ** are the two shorter sides (it doesn't matter which)

then Pythagoras' Theorem says that:

Does a triangle contain a right angle?
If a triangle is a right-angled triangle, the sides will obey Pythagoras' Theorem. {square and add the two shorter sides should give the same result as squaring the hypotenuse}


 * Example **

Decide if each of these triangles are right-angled triangles
 * 1) Side Lengths : 5.5cm, 11.4cm, 13.2cm
 * 2) Side Lengths : 9cm, 12cm, 15cm

__**Solution: (a)**__

Side lengths : 5.5cm, 11.4cm, 13.2cm {square and add the two shorter lengths} math . \qquad \qquad 5.5^2 + 11.4^2 = 160.21 math

{square the longer side} math . \qquad \qquad 13.2^2=174.24 math

{compare the two results} math . \qquad \qquad 160.21 \neq 174.24 math

Results are __**not**__ equal so triangle (a) is __**not**__ a right angled triangle.

__**Solution: (b)**__

Side lengths : 9cm, 12cm, 15cm {square and add the two shorter lengths} math . \qquad \qquad 9^2 + 12^2 = 225 math

{square the longer side} math . \qquad \qquad 15^2=225 math

{compare the two results} math . \qquad \qquad 225 = 225 math

Results __**are**__ equal so triangle (b) __**is**__ a right angled triangle.

Finding Side Lengths
We can use Pythagoras' Theorem to find the length of one side of a right angled triangle if we know the lengths of the other two sides.

Finding the Hypotenuse
To find the length of the hypotenuse
 * square the other two sides then **ADD**
 * take the square root of the result
 * answer is a length so include units (when available)


 * Example 1 **

For the triangle shown, calculate the length of the hypotenuse (x) correct to 1 decimal place. __**Solution:**__

{Label the sides of the triangle with a, b, c} math \\ . \qquad c^2 = a^2 + b^2 \qquad \{ \textit{Sub } a = 4 \textit{ and } b = 7 \} \\ \\ . \qquad x^2 = 4^2 + 7^2 \\ \\ . \qquad x^2 = 16 + 49 \\ \\ . \qquad x^2 = 65 math

{Find x by taking the square root:} math \\ . \qquad x = \sqrt{65} \\ \\ . \qquad x = 8.06225 ... \\ \\ . \qquad x = 8.1 \; \text{cm} \qquad \{ \textit{rounded to 1 decimal place} \} math

Finding a Shorter Side
Pythagoras' Theorem can be rearranged, so math \\ . \qquad c^2 = a^2 + b^2 \qquad \{ -b^2 \} \\ \\ . \qquad a^2 = c^2 - b^2 math

Therefore, to find the length of one of the shorter sides
 * square the other two sides then **SUBTRACT**
 * take the square root of the result
 * answer is a length so include units (when available)


 * Example 2 **

For the triangle shown, calculate the length of the unknown side (x) correct to 1 decimal place. __**Solution:**__

{Label the sides of the triangle with a, b, c} math \\ . \qquad a^2 = c^2 - b^2 \qquad \{ \textit{Sub } c = 14 \textit{ and } b = 8 \} \\ \\ . \qquad x^2 = 14^2 - 8^2 \\ \\ . \qquad x^2 = 196 - 64 \\ \\ . \qquad x^2 = 132 math

{Find x by taking the square root:} math \\ . \qquad x = \sqrt{132} \\ \\ . \qquad x = 11.4891 ... \\ \\ . \qquad x = 11.5 \; \text{cm} \qquad \{ \textit{rounded to 1 decimal place} \} math


 * Common Errors are: **
 * forgetting to take the square root (stopping too early)
 * mixing up between finding the hypotenuse (ADD) and finding a shorter side (SUBTRACT)
 * not rounding off to the specified number of decimal places (or rounding off incorrectly)
 * not supplying the units for the answer (m or cm etc)
 * getting the notation wrong (when to write "x 2 =" and when to write "x =" )

Word Problems

 * Draw a diagram of the situation.
 * Identify the right angled triangle within the diagram
 * use Pythagoras' Theorem to find the unknown side
 * Don't forget to
 * answer the question that was asked
 * include units in your answer


 * Example 3 **

A 4.5m ladder leans up against a vertical wall. The foot of the ladder is 1.2m from the base of the wall. How far up the wall does the ladder reach? Give your answer to 1 decimal place.

__**Solution:**__ Draw Diagram

The diagram shows that we are finding a shorter side,
 * so square and then **SUBTRACT**

math \\ . \qquad a^2 = c^2 - b^2 \qquad \{ \textit{Sub } c = 4.5 \textit{ and } b = 1.2 \} \\ \\ . \qquad a^2 = 4.5^2 - 1.2^2 \\ \\ . \qquad a^2 = 18.81 math

{Find a by taking the square root:} math \\ . \qquad a = \sqrt{18.81} \\ \\ . \qquad a = 4.3 \; \text{m} \qquad \{ \textit{rounded to 1 decimal place} \} math

The ladder reaches 4.3m up the wall.

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