Points M and N are on the sides PQ and QR respectively of a triangle PQR, right angled at Q. If PN = 9 cm, MR = 7 cm, and MN = 3 cm, then find the length of PR (in cm).
11
Given:
Points M and N are on the sides PQ and QR respectively of ΔPQR.
PN = 9 cm
MR = 7 cm
MN = 3 cm
Concept used:
Pythagoras theorem
Calculation:
Let the measure of the side PM, MQ, QN, NR be d, a, b and c respectively.
From ΔPQR, (a + d)2 + (b + c)2 = PR2
So, ΔPQN, ΔMQN & ΔMQR are all right-angled triangles.
Now,
from ΔPQN,
(a + d)2 + b2 = 92
⇒ (a + d)2 + b2 = 81 ....(1)
from ΔMQN,
a2 + b2 = 32
⇒ a2 + b2 = 9 ....(2)
from ΔMQR,
a2 + (b + c)2 = 72
⇒ a2 + (b + c)2 = 49 ....(3)
(1) + (3),
(a + d)2 + b2 + a2 + (b + c)2 = 81 + 49
⇒ (a + d)2 + (b + c)2 + (b2 + a2) = 130
⇒ (a + d)2 + (b + c)2 + 9 = 130
⇒ (a + d)2 + (b + c)2 = 121
⇒ PR2 = 121
⇒ PR = 11
∴ The length of PR is 11 cm.
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