Page 61 - Mathematics Class - X

5. PQRS is a trapezium.
6. The trapezium PQRS shown in Fig. (d) is divided into three triangles: PAS, AQR and SAR.

OBSERVATIONS
From Fig (d), by actual measurement, we have

PA = ____________ PS = ____________ SA = ____________
QA = ____________ QR = ____________ AR = ____________

\ PS + PA = ____________ SA = ____________
2
2
2
\ QA + QR = ____________ AR = ____________
2
2
2
Thus, a + b = ____________
2
2
From Fig. (d), we have
1. ∆SAR is right angled at A.
2. Area of ∆APS = 1 ba square units
2
1
Area of ∆AQR = ab square units
2
1
Area of ∆SAR = c square units
2
2
3. Area of trapezium PQRS = area (∆APS) + area (∆AQR) + area (∆SAR)

So, 1 (a + b)(a + b) = ( 1 ab) + ( 1 ab) + 1 c = 1 (ab + ab + c 2 )
2
2 2 2 2 2
(a + b) = (ab + ab + c ) [Q (a + b) (a + b) = (a + b) ]
2
2
2
a + b + 2ab = (ab + ab + c )
2
2
2
a + b + 2ab = 2ab + c 2
2
2
a + b = c 2
2
2
Hence, Pythagoras theorem is verified.
INFERENCE
We have verified the Pythagoras theorem.

EXTENDED TASK
1. Prove Pythagoras theorem with any other method. There are almost 99 ways in which this theorem can be
proved. Try it!
2. Make a computer program to find the various Pythagorean triplets.

APPLICATION
Using Pythagoras theorem, we can find out the third side of a right-angled triangle whose two other sides
are given.

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