Calculating the length of one of the shorter sides
Rearrange 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. to calculate the length of one of the shorter sides.
\(a^2 + b^2 = c^2\) calculates the length of the longest side.
To calculate the length of one of the shorter sides, rearrange the formula to make \(a^2\) or \(b^2\) the subject.
\(a^2 = c^2 - b^2\) or \(b^2 = c^2 - a^2\)
Then take the square root to calculate the length \(a\) or \(b\).
Example
Calculate the length AB.
\(a^2 + b^2 = c^2\)
\(8^2 + b^2 = 10^2\)
Rearrange the formula to make \(b^2\) the subject:
\(b^2 = 10^2 - 8^2\)
\(b^2 = 36\)
\(b = \sqrt{36}\)
\(b = 6~\text{cm}\)
The length AB is 6 cm.
Question
Calculate the length AO. Give the answer to one decimal place.
\(a^2 + b^2 = c^2\)
\(3^2 + b^2 = (\sqrt{17})^2\)
Rearrange the formula to make \(b^2\) the subject.
\(b^2 = (\sqrt{17})^2 - 3^2\)
\(b^2 = 17 - 9\)
\(b^2 = 8\)
\(b = \sqrt{8}\)
\(b = 2.8~\text{cm}\)
The length AO is 2.8 cm.