Solving problems using Pythagoras' theorem
Pythagoras's theorem applies to right-angled triangles. The area of the square drawn on the hypotenuse is equal to the sum of the squares drawn on the other two sides. can be used to solve Having only two dimensions, usually length (or height) and width. problems which involve calculating the length of a right-angled triangle.
Example
Calculate the length UQ. Give the answer to one decimal place.
Draw the right-angled triangle URQ and label the sides.
\(a^2 + b^2 = c^2\)
\(4^2 + 10^2 = c^2\)
\(116 = c^2\)
\(c = \sqrt{116}\)
\(c = 10.8~\text{cm}\)
The length UQ is 10.8 cm.
It may be necessary to use Pythagoras' theorem more than once in a problem.
Question
Calculate the length S. Give the answer to one decimal place.
It is necessary to calculate the perpendicular height of the triangles first before calculating S.
First, draw the smaller triangle and label the sides.
\(a^2 + b^2 = c^2\)
\(3^2 + b^2 = 7^2\)
Rearrange the formula to make \(b^2\) the subject.
\(b^2 = 7^2 - 3^2\)
\(b^2 = 40\)
\(b = \sqrt{40}\)
\(\sqrt{40}\) is a surd. Do not round this answer yet.
Draw the larger triangle and label the sides.
\(a^2 + b^2 = c^2\)
\((\sqrt{40})^2 + 6^2 = s^2\)
\(s^2 = 76\)
\(s = \sqrt{76}\)
\(s = 8.7~\text{cm}\)