Page 2 - LN

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Prove that in a right-angled triangle, the square of the hypotenuse is equal to
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the sum of the squares of the other two sides..
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P
Q
Given: A right-angled PQR in which Q = 90
2
2
P
+
To prove: PR = QR + PQQ 2

Construction: Draw QM PR.
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P
w
Q
a

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v
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Proof: In PMQ and PQR, we have,,

,
R
P = P (common)
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9
a
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ch
(
e
PMQ = PQR (each is 900 )

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Therefore, by AA Criterion of similarity, we have,,
a
PMQ ~ PQR
Therefore,

(i)
e
P
,

Similarly in RMQ and RQP, we have,,
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Q
R
w
a

ve
R = R (common)
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9
a
(
i
RMQ = RQP (each is 900 )

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e
ch

A
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f
T
Therefore, by AA similarity, we have,,
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h
y
b
A

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r

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m
i
RMQ ~ RQP
Therefore,

(ii)
Adding (i) and (ii), we get,
2
2
R
M
M

PQ + RQ = PM. PR + RM. RPP
R
.
+
R
=
P

.
P
(

= RP (PM + RM))
M
P
+

R
M

=

R
P
= RP RP
= RP 2
2
2
2
Hence, PQ + QR = PR

2