11.01 Pythagoras' theorem

Pythagoras' theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Referring to the diagram, the theorem can be written algebraically: a^2 + b^2 = c^2where c represents the length of the hypotenuse and a, b are the lengths of the two shorter sides.

So if two sides of a right-angled triangle are known and one side is unknown, this relationship can be used to find the length of the unknown side.

The equation can be rearranged to make any unknown side length of the triangle the subject of the formula:c=\sqrt{a^2+b^2} \qquad \qquad b=\sqrt{c^2-a^2} \qquad\qquad a=\sqrt{c^2-b^2}

Examples

Example 1

Calculate the value of c in the triangle below.

Worked Solution

Create a strategy

Use Pythagoras' Theorem: c^2=a^2+b^2.

Apply the idea

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use Pythagoras' Theorem
\displaystyle c^2	\displaystyle =	\displaystyle 14^2+48^2	Substitute a=14, \, b=48
\displaystyle c^2	\displaystyle =	\displaystyle 2500	Evaluate
\displaystyle c	\displaystyle =	\displaystyle \sqrt{2500}	Square root both sides
\displaystyle c	\displaystyle =	\displaystyle 50\text{ cm}	Evaluate

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Example 2

Calculate the value of b in the triangle below. Give your answer correct to two decimal places.

Worked Solution

Create a strategy

Use Pythagoras' Theorem: c^2=a^2+b^2.

Apply the idea

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use Pythagoras' Theorem
\displaystyle 24^2	\displaystyle =	\displaystyle 10^2+b^2	Substitute c=24 and a=10
\displaystyle 576	\displaystyle =	\displaystyle 100+b^2	Evaluate the powers
\displaystyle 476	\displaystyle =	\displaystyle b^2	Subtract 100 from both sides
\displaystyle b	\displaystyle =	\displaystyle \sqrt{476}	Square root both sides
\displaystyle b	\displaystyle =	\displaystyle 21.82	Evaluate and round

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To apply the Pythagorean theorem to real-life situations, we can follow these four simple steps.

1. Look for right-angled triangles in the scenario

2. Choose which side, hypotenuse or a shorter side, you are trying to find

3. Find the lengths of the other two sides

4. Apply the relevant formula and substitute the lengths of the other two sides

Examples

Example 3

The screen on a handheld device has dimensions 9 \text{ cm} by 5\text{ cm}, and a diagonal of length x cm.

Find the value of x, correct to two decimal places.

Worked Solution

Create a strategy

Use the Pythagoras' theorem: a^{2}+b^{2}=c^{2}

Apply the idea

\displaystyle x^{2}	\displaystyle =	\displaystyle 9^{2}+5^{2}	Substitute a=9,\,b=5,\,and c=x
\displaystyle x^{2}	\displaystyle =	\displaystyle 81+25	Evaluate the squares
\displaystyle x^{2}	\displaystyle =	\displaystyle 106	Evaluate the sum
\displaystyle x	\displaystyle =	\displaystyle \sqrt{106}	Take the square root of both sides
\displaystyle x	\displaystyle =	\displaystyle 10.30\text{ cm}	Simplify and round to two decimal places

Example 4

VUTR is a rhombus with perimeter 160 \text{ cm}. The length of diagonal RU is 46 \text{ cm}.

a

Find the length of VR.

Worked Solution

Create a strategy

Use the fact that all sides in a rhombus are equal.

Apply the idea

To get the length of one side, divide the perimeter by 4.

\displaystyle VR	\displaystyle =	\displaystyle \dfrac{160}{4}	Divide 160 by 4
	\displaystyle =	\displaystyle 40 \text{ cm}	Evaluate

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b

Find the length of RW.

Worked Solution

Create a strategy

Recall that the diagonals of a rhombus bisect each others.

Apply the idea

Since the diagonals bisect each other RW=\dfrac{1}{2}RU.

\displaystyle RW	\displaystyle =	\displaystyle \dfrac{RU}{2}	Halve RU
	\displaystyle =	\displaystyle \dfrac{46}{2}	Subsitute 46
	\displaystyle =	\displaystyle 23 \text{ cm}	Evaluate

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c

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

Worked Solution

Create a strategy

Use the fact that \triangle VRW is right-angled, so the unknown side x can be found using Pythagoras' theorem given by c^2=a^2+b^2.

Apply the idea

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Write the formula
\displaystyle RV^2	\displaystyle =	\displaystyle RW^2+x^2	Substitute a=RW,\,b=x,\,c=RV
\displaystyle 40^2	\displaystyle =	\displaystyle 23^2+x^2	Substitute RW=23,\,RV=40
\displaystyle x^2	\displaystyle =	\displaystyle 40^2-23^2	Subtract 23^2 from both sides
\displaystyle x^2	\displaystyle =	\displaystyle 1071	Evaluate the right side
\displaystyle \sqrt{x^2}	\displaystyle =	\displaystyle \sqrt{1071}	Take the square root of both sides
\displaystyle x	\displaystyle =	\displaystyle 32.73 \text{ cm}	Evaluate

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d

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

Worked Solution

Create a strategy

Recall that the diagonals of a rhombus bisect each others.

Apply the idea

Since VT \perp RU, this means VT=2VW.

Solving for the length of VT, we have:

\displaystyle VT	\displaystyle =	\displaystyle 2VW	Write the formula
	\displaystyle =	\displaystyle 2\times 32.73	Subsitute 32.73 for VW
	\displaystyle =	\displaystyle 65.46 \text{ cm}	Evaluate

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