The square of length of hypotenuse equals to sum of squares of lengths of opposite side and adjacent side in a right angled triangle is called Pythagorean Theorem.

It is also called as Pythagoras theorem. It describes how hypotenuse maintains a relation with remaining two sides of the right angled triangle. It is very important property of the right angled triangle and often used in dealing the triangles.

**(Length of Hypotenuse) ^{2}** =

A right angled triangle is required to prove Pythagorean Theorem. So, construct a right angled triangle. In this example, a right angled triangle $PQR$ is constructed in following three steps.

- Draw a $16$ centimetres horizontal line segment and assume to call it line segment $\stackrel{\u203e}{PQ}$.
- Draw a $12$ centimetres vertical line segment from point $P$ and assume its endpoint is point $R$.
- Join points $R$ and $Q$ by a line segment and measure the length of the line segment $\stackrel{\u203e}{RQ}$. The length of line segment $\stackrel{\u203e}{RQ}$ is measured $20$ centimetres exactly.

Thus, the right angled triangle $PQR$ is constructed geometrically. The line segments $\stackrel{\u203e}{PQ},\stackrel{\u203e}{PR}$ and $\stackrel{\u203e}{RQ}$ are known as adjacent side, opposite side and hypotenuse of the $\Delta PQR$.

Calculate the ratio of length of the opposite side to length of the hypotenuse.

$\frac{Length\; of\; the\; opposite\; side}{Length\; of\; the\; hypotenuse}=\frac{PR}{RQ}$

$\frac{PR}{RQ}=\frac{12}{20}=0.6$

Calculate the ratio of length of the adjacent side to length of the hypotenuse.

$\frac{Length\; of\; the\; adjacent\; side}{Length\; of\; the\; hypotenuse}=\frac{PQ}{RQ}$

$\frac{PQ}{RQ}=\frac{16}{20}=0.8$

Draw a line from point $P$ on to hypotenuse $\stackrel{\u203e}{RQ}$ but remember, the line should be perpendicular to the hypotenuse. Assume, the line insects the hypotenuse at point $S$. Therefore, $\stackrel{\u203e}{PS}\perp \stackrel{\u203e}{RQ}$.

- Measure the length of the line segment $\stackrel{\u203e}{RS}$ and it is measured as $7.2$ centimetres.
- Measure the length of the line segment $\stackrel{\u203e}{SQ}$ and it is measured as $12.8$ centimetres.

Consider the $\Delta RPS$.

Calculate the ratio of length of the opposite side to length of hypotenuse.

$\frac{Length\; of\; the\; opposite\; side}{Length\; of\; the\; hypotenuse}=\frac{RS}{PR}$

$\frac{RS}{PR}=\frac{7.2}{12}=0.6$

As you know that the ratio of length of the opposite side to length of the hypotenuse is also $0.6$ in right angled triangle $PQR$ because the right angled triangles $\Delta PQR$ and $\Delta RPS$ are similar triangles.

Therefore $\frac{PR}{RQ}=0.6$ and $\frac{RS}{PR}=0.6$

So, $\frac{PR}{RQ}=\frac{RS}{PR}$

${PR}^{2}=RQ\times RS$

Consider the $\Delta SQP$.

Calculate the ratio of length of the adjacent side to length of hypotenuse.

$\frac{Length\; of\; the\; adjacent\; side}{Length\; of\; the\; hypotenuse}=\frac{SQ}{PQ}$

$\frac{SQ}{PQ}=\frac{12.8}{16}=0.8$

As you also know that ratio of length of the adjacent side to length of the hypotenuse is also $0.8$ in right angled triangle $PQR$ because the right angled triangles $\Delta PQR$ and $\Delta SQP$ are similar triangles.

Therefore $\frac{PQ}{RQ}=0.8$ and $\frac{SQ}{PQ}=0.8$

So, $\frac{PQ}{RQ}=\frac{SQ}{PQ}$

${PQ}^{2}=RQ\times SQ$

According to $\Delta PQR$ and $\Delta RPS$

${PR}^{2}=RQ\times RS$

According to $\Delta PQR$ and $\Delta SQP$

${PQ}^{2}=RQ\times SQ$

Now, add these two expressions.

${PR}^{2}+{PQ}^{2}=RQ\times RS+RQ\times SQ$

$\Rightarrow {PR}^{2}+{PQ}^{2}=RQ\times (RS+SQ)$

The perpendicular line $\stackrel{\u203e}{PS}$ is split the hypotenuse $\stackrel{\u203e}{RQ}$ as sides $\stackrel{\u203e}{RS}$ and $\stackrel{\u203e}{SQ}$. Therefore, $RS+SQ=RQ$

$\Rightarrow {PR}^{2}+{PQ}^{2}=RQ\times RQ$

$\Rightarrow {PR}^{2}+{PQ}^{2}={RQ}^{2}$

According to right angled triangle $PQR$

- $PR$ is length of the opposite side
- $PQ$ is length of the adjacent side
- $RQ$ is length of the hypotenuse

The square of length of the hypotenuse is equal to the sum of square of length of the opposite side and square of length of the adjacent side.

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