Pythagoras Theorem

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   =   (Length of Opposite side)2   +   (Length of Adjacent side)2

Proof

right angled triangle

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.

  1. Draw a 16 centimetres horizontal line segment and assume to call it line segment PQ.
  2. Draw a 12 centimetres vertical line segment from point P and assume its endpoint is point R.
  3. Join points R and Q by a line segment and measure the length of the line segment RQ. The length of line segment RQ is measured 20 centimetres exactly.

Thus, the right angled triangle PQR is constructed geometrically. The line segments PQ, PR and RQ are known as adjacent side, opposite side and hypotenuse of the ΔPQR.

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

Length of the opposite sideLength of the hypotenuse  =   PRRQ

PRRQ = 1220 = 0.6

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

Length of the adjacent sideLength of the hypotenuse  =   PQRQ

PQRQ = 1620 = 0.8

similar right angled triangles

Draw a line from point P on to hypotenuse RQ but remember, the line should be perpendicular to the hypotenuse. Assume, the line insects the hypotenuse at point S. Therefore, PS RQ.

  1. Measure the length of the line segment RS and it is measured as 7.2 centimetres.
  2. Measure the length of the line segment SQ and it is measured as 12.8 centimetres.

Consider the ΔRPS.

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

Length of the opposite sideLength of the hypotenuse  =   RSPR

RSPR = 7.212 = 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 ΔPQR and ΔRPS are similar triangles.

Therefore PRRQ = 0.6 and RSPR = 0.6

So, PRRQ  =   RSPR

PR2 = RQ × RS

Consider the ΔSQP.

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

Length of the adjacent sideLength of the hypotenuse  =   SQPQ

SQPQ = 12.816 = 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 ΔPQR and ΔSQP are similar triangles.

Therefore PQRQ = 0.8 and SQPQ = 0.8

So, PQRQ  =   SQPQ

PQ2 = RQ × SQ

According to ΔPQR and ΔRPS

PR2 = RQ × RS

According to ΔPQR and ΔSQP

PQ2 = RQ × SQ

Now, add these two expressions.

PR2 + PQ2 = RQ × RS + RQ × SQ

⇒   PR2 + PQ2 = RQ × (RS + SQ)

The perpendicular line PS is split the hypotenuse RQ as sides RS and SQ. Therefore, RS + SQ = RQ

⇒   PR2 + PQ2 = RQ × RQ

⇒   PR2 + PQ2 = RQ2

According to right angled triangle PQR

  1. PR is length of the opposite side
  2. PQ is length of the adjacent side
  3. 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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