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Pythagoras theorem

Pythagoras' Theorem (also called the Pythagorean theorem) has been used from antiquity to modern times to solve practical problems in situations from surveying, construction and design that involve right-angled triangles, such as connected beams, posts and rafters in house roofs.

Use this page to revise the following concepts within Pythagoras' theorem:

Defining Pythagoras' Theorem
Applying the theorem

Defining Pythagoras' Theorem

Pythagoras' Theorem states that for a right-angled triangle, the sum of the areas of the squares on the two shorter sides (called the legs of the triangle), is equal to the area of the square of the longest side (called the hypotenuse of the triangle). The theorem says that:

\[ a^2 + b^2 = c^2 \]

Where:

\(a\) and \(b\) are the legs of a right-angled triangle
\(c\) is the hypotenuse of a right-angled triangle

Note that the hypotenuse is the side opposite the right angle. The basis of Pythagoras' theorem relates the measurement of the area of squares with the side lengths of a right-angled triangle, as illustrated in the following diagram:

Pythagoras' theorem provides the basis for:

calculating lengths of sides of right-angled triangles
calculating the distance between points in the Cartesian coordinate plane.

Applying the theorem

The application of Pythagoras' theorem has the following steps:

Draw a right-angled triangle showing the two known lengths and the length to be determined
Substitute the known values into the equation \(a^2 + b^2 = c^2\)
Solve the equation for the unknown value, accepting only positive square root values.
For a practical problem, approximate the answer to an accuracy suitable for the context

Worked Example
A right-angled triangle has two shorter side lengths of 9 and 17 respectively. The length of the longest side can be calculated as follows:
Using Pythagoras' theorem
Substituting into the formula:
\[9^2 + 17^2 = c^2\]
Evaluating:
\[\begin{align} c^2 &= 81 + 289 \\ &=370 \\ c&=\sqrt{370} \end{align} \]
Worked Example
A right-angled triangle has a hypotenuse of length 24 and one shorter side of length 13. The length of the other shorter side can be calculated as follows:
Using Pythagoras' theorem
Substituting into the formula:
\[13^2 + b^2 =24^2\]
Evaluating:
\[\begin{align} 169 + b^2 &= 576\\ b^2 &= 407\\ b&= \sqrt{407} \end{align}\]
The distance between two points in the Cartesian coordinate plane with coordinates \((x_1,y_1)\) and \((x_2,y_2)\) respectively, can be determined by:
- identifying lengths in the corresponding right-angled triangle and applying Pythagoras' theorem, or
- applying the distance formula
The distance formula is obtained by applying Pythagoras' theorem to the diagram:
Using Pythagoras' theorem, this gives:
\[ d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \]
Hence, the distance between two points \((d)\), or length of a line segment, joining two points on the cartesian plane can be given by the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Worked Example
Find the distance between the two points in the Cartesian coordinate plane with coordinates \((-2,1)\) and \((5,4)\).
The two points are shown on the graph below:
Using Pythagoras' theorem directly, the distance between the two points is:
\[
d = \sqrt{7^2 + 3^2}
\]
\[
d = \sqrt{49 + 9}
\]
\[
d = \sqrt{58} \approx 7.6
\]
Alternatively, using the distance formula with \((x_1,y_1) = (-2,1)\) and \((x_2,y_2) = (5,4)\):
\[ d = \sqrt{(5 - (-2))^2 + (4 - 1)^2} \]
\[
d = \sqrt{7^2 + 3^2}
\]
\[
d = \sqrt{49 + 9}
\]
\[
d = \sqrt{58} \approx 7.6
\]