# Solving problems using Pythagoras' theorem

Question

A cable is attached, 30 metres above ground level, to a post.

The other end of the cable is 14 metres from the base of the post.

Regulations state that the cable must measure less than 35 metres.

Are the regulations being met?

${x^2} = {30^2} + {14^2}$

${x^2} = 900 + 196$

${x^2} = 1096$

$x = \sqrt {1096}$

$x = 33.1\,(to\,1\,d.p.)$

Yes, regulations are being met because 33.1m is less than 35m.

## Using Pythagoras with coordinates

We can also use Pythagoras to find the distance between two points.

Question

If A has coordinates (3, 4) and B has coordinates (10, 12), calculate the length of AB. Give your answer to one decimal place.

Plot the coordinates onto a graph.

Join up A and B to show the line AB then form a right-angled triangle.

${a^2} = {b^2} + {c^2}$

${a^2} = {8^2} + {7^2}$

${a^2} = 64 + 49$

${a^2} = 113$

$a = \sqrt {113}$

$a = 10.6\,units\,(to\,1\,d.p.)$

The length of AB is 10.6 units.

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