PQR is a right-angled triangle with the right angle at Q and k being the length of the perpendicular from Q on PR. If l, m and n are the lengths of sides PQ, QR, and PR respectively, then which of the following holds true?
1
k
2
=
1
l
2
−
1
m
2
1
k
2
=
1
l
2
+
1
m
2
1
k
2
=
1
m
2
−
1
l
2
1
m
2
=
1
l
2
−
2
k
2
Answers
Answer:
Area of circle with QR as diameter =
2
π(5
2
)
=
2
25π
cm
2
Area of semi circle with diameter PR
=
2
π(4
2
)
=8π cm
2
Area of semi circle with PQ diameter
=
2
π(3
2
)
=
2
9π
cm
2
Area of triangle PQR=
2
1
×6×8=24 cm
2
Area of shaded regions
=(8π+
2
9π
+24)−(
2
25π
)
=12.5π+24−12.5π
=24 cm
2
Given:
Δ PQR with ∠ Q = 90°
The length of the perpendicular from Q on PR = k
Length of PQ= l
Length of QR= m
Length of PR= n
To Find:
Relation between sides of the triangle
Solution:
Area of a triangle = 1/2 X Base X Height
Area of Δ PQR = (1/2) X PQ X QR
= 1/2 X l X m - (1)
Since given that k is a perpendicular from Q on PR
So, the area of Δ PQR can also be expressed as (1/2) X PR X k
1/2 n X k -(2)
Sice both the equations give the area of the same triangle, we can conclude that (1) = (2)
or 1/2 X l X m = 1/2 X n X k
=> lm = nk
Hence, the option (c) l.m = n.k is the correct relation.
(The complete question is: PQR is a right-angled triangle with the right angle at Q and k being the length of the perpendicular from Q on PR. If l, m and n are the lengths of sides PQ, QR, and PR respectively, then which of the following holds true?
a) k² = l²
b) m² = 2k
c) lm = nk
d) lk = nm)