Pythagoras Questions | Worksheets and Revision | MME

Pythagoras Questions, Worksheets and Revision

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What you need to know

Pythagoras

Pythagoras’ theorem is an equation that describes a relationship between the 3 sides of a right-angled triangle. We can use it to determine a missing length when given two other lengths. 

Being able to rearrange equations will help with this topic. 

Pythagoras’ Theorem

The equation is

a^2+b^2=c^2

where c is always the hypotenuse (the longest side), and a and b are the two other sides.

Example 1: Finding a Length

Find the length of the side marked x on the right angled triangle shown.

[2 marks]

In order to find this using Pythagoras’ theorem, we need to work out which side corresponds to each of the letters \textcolor{red}{a}, \textcolor{limegreen}{b}, and \textcolor{blue}{c} in the equation.

\textcolor{blue}{c} = \textcolor{blue}{10} cm

\textcolor{limegreen}{b} = \textcolor{limegreen}{6} cm

\textcolor{red}{a} = \textcolor{red}{x}

Then, the equation a^2+b^2=c^2 becomes

\textcolor{red}{x}^2+\textcolor{limegreen}{6}^2=\textcolor{blue}{10}^2

\textcolor{red}{x}^2+\textcolor{limegreen}{36}=\textcolor{blue}{100}

Now we can solve for \textcolor{red}{x}

\textcolor{red}{x}^2=\textcolor{blue}{100}-\textcolor{limegreen}{36}=64.

\textcolor{red}{x}=\sqrt{64}=8\text{cm}.

 

Example 2: Calculating Length of a Line

Calculate the distance between (-4, -1) and (3, 4).

Give your answer to 1dp.

[2 marks]

First we plot the two points on a pair of axes, draw a line between them, and then draw a triangle underneath

Side 1 = 4 -(-1) = \textcolor{red}{5}

Side 2 = 3 -(-4) = \textcolor{limegreen}{7}

next we input these lengths into Pythagoras theorem.

\textcolor{red}{a^2}+\textcolor{limegreen}{b^2}=\textcolor{blue}{c}^2

\textcolor{blue}{c}^2=\textcolor{red}{5^2}+\textcolor{limegreen}{7^2}=\textcolor{red}{25}+\textcolor{limegreen}{49}=74

\textcolor{blue}{c}=\sqrt{74}=8.6 1dp

Note: The length you are trying to calculate will always be the hypotenuse in these examples. 

Example Questions

The missing side is the longest side, thus the equation a^2+b^2=c^2 becomes:

 

8^2+14^2=c^2

 

Evaluating the two squares, we get,

 

c^2=64+196=260

 

Then, square rooting both sides of this equation, we get,

 

BC=\sqrt{260}=16.1\text{ cm (1 d.p.)}

To do this, we will plot the points on a graph, draw a line between them, and construct a right-angled with one side parallel to the x axis and another parallel to y axis. The resulting picture looks like:

 

 

We can see that the distance between the two points is the longest side of a right-angle triangle, so the equation a^2+b^2=c^2 becomes,

 

10^2+3^2=c^2

 

Evaluating the squares, we get

 

c^2=100+9=109

 

Finally, square rooting both sides, we get,

 

c=\sqrt{109}=10.4\text{ (3 s.f.)}

The missing side is the hypotenuse, thus the equation a^2+b^2=c^2 becomes:

 

5.9^2+6.7^2=c^2

 

Evaluating the two squares, we get,

 

c^2=34.81+44.89=79.7

 

Then, square rooting both sides of this equation, we get,

 

QR=\sqrt{79.7}=8.9\text{ cm (1 d.p.)}

Applying a rearranged form of Pythagoras, letting, a, be the missing length we want to find, then a^2=c^2-b^2 becomes, 

 

a^2=5.1^2-3.1^2

 

Evaluating the two squares, we get,

 

a^2=26.01-9.61=16.4

 

Then, square rooting both sides of this equation, we get,

 

a=\sqrt{16.4}=4.0\text{ cm (1 d.p.)}

Applying a rearranged form of Pythagoras, letting, a, be the length of the wall we want to find, then a^2=c^2-b^2 becomes, 

 

a^2=2.9^2-1.3^2

 

Evaluating the two squares, we get,

 

a^2=8.41-1.69=6.72

 

Then, square rooting both sides of this equation, we get,

 

a=\sqrt{6.72}=2.6\text{ cm (1 d.p.)}

Worksheets and Exam Questions

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(NEW) Pythagoras Exam Style Questions - MME

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Pythagoras 1 - Drill Questions

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Pythagoras 2 - Drill Questions

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