Key points
An understanding of how to use Pythagorasโ theorem to find missing sides in a right-angled triangle is essential for applying the theorem in different contexts.
Pythagorasโ theorem can be used to find the distance between two points. This is done by joining the points together to form the hypotenuseThe longest side of a right-angled triangle, which is always opposite the right angle. When labelling a length as the hypotenuse, it can be shortened to ๐. of a right-angled triangle and using the theorem \(a\)ยฒ + \(b\)ยฒ = \(c\)ยฒ to find the length of the hypotenuse.
Pythagorasโ theorem can also be used to find missing lengths in shapes, such as rectangles and isosceles triangleA triangle with two equal sides. This means two angles are equal. once the shape has been split into right-angled triangles.
How to find the distance between two points
Follow these steps to calculate the distance between two coordinatesThe ordered pair of numbers (๐, ๐) that defines the position of a point is the coordinate pair (or the coordinates.) on a set of axesTwo reference lines, one horizontal and one vertical, that cross at right-angles. They are used to define the position of a point on a grid. Axes is the plural of axis. . The same steps can be taken to calculate the length of a line segmentA specific part of a line between two points. where the two end points are the two coordinates.
- Join the two coordinates with a straight line. This will become the hypotenuseThe longest side of a right-angled triangle, which is always opposite the right angle. When labelling a length as the hypotenuse, it can be shortened to ๐. of a right-angled triangle and is the length that needs to be found.
- Draw a horizontalA line that is parallel to the horizon. and verticalA line that is perpendicular to the horizon. line from the two coordinates to form a right-angled triangle.
- Calculate the horizontal length of the triangle by finding the difference between the x-coordinates. This is side a of the right-angled triangle.
- Calculate the vertical length of the triangle by finding the difference between the y-coordinates. This is side b of the right-angled triangle.
- Substitute the values of \(a\) and \(b\) into Pythagorasโ theorem: \(a\)ยฒ + \(b\)ยฒ = \(c\)ยฒ.
- Add the squares together, then find the square root to calculate the c, the hypotenuse. The hypotenuse is the distance between the two points.
Example
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Question
Find the distance between points R and S.
Draw a horizontal and vertical line from the two coordinates to form a right-angled triangle.The width of the triangle is 2
This can be labelled \(a\).
The height of the triangle is 4
This can be labelled \(b\).
The line segment RS is labelled \(c\).
Substitute the values of a, b and c into Pythagorasโ theorem to give 2ยฒ + 4ยฒ = \(c\)ยฒ.
4 + 16 = 20
To find the value of \(c\), calculate the square root of 20
This gives the final answer of \(c\)ยฒ = 4.4721โฆ which rounded to one decimal place = 4.5
Therefore, the distance between R and S is 4.5
Applying Pythagorasโ theorem to other shapes
Pythagorasโ theorem can be used to find the diagonal of a rectangle. The width and height of the rectangle become \(a\) and \(b\) in the formula and \(c\) is the diagonal length.
\(a\)ยฒ + \(b\)ยฒ = \(c\)ยฒ
isosceles triangleA triangle with two equal sides. This means two angles are equal. can be split into two right-angled triangles. This means Pythagorasโ theorem can be used to find the lengths of missing sides, such as the perpendicularPerpendicular lines are at 90ยฐ (right angles) to each other. height. It is important to remember that when an isosceles triangle is cut in half this way, the base length is also cut in half.
Examples
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Question
A question asks to calculate the height of the isosceles triangle, \(b\). The start of the working is shown on the right.
What mistake has been made?
The lengths of the isosceles triangle have been used in the equation. However, Pythagorasโ theorem only works for right-angled triangles.
The isosceles triangle needs to be split into two right-angled triangles. The width of the right-angled triangle is 12 cm. 12 cm is the correct value for \(a\) (not 24 cm).
The correct equation is 12ยฒ + \(b\)ยฒ = 37ยฒ.
Visual proof of Pythagoras' theorem
Explore a visual way of showing Pythagorasโ theorem below.
Example
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Practise using Pythagoras' theorem
Quiz
Practise calculating different lengths of sides using Pythagoras' theorem with this quiz. You may need a pen and paper to help you with your answers.
Real-life maths
Navigation systems on ships use Pythagorasโ theorem to calculate the shortest distance to a certain destination.
Knowing the shipโs coordinates and the coordinates of their destination means they can find the distance between those two points.
It is vital to know this distance to check if the ship has enough fuel to get to its destination safely.
Game - Divided Islands
Divided Islands. gameDivided Islands
Use your maths skills to help the islanders of Ichi build bridges and bring light back to the islands in this free game from BBC Bitesize.
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