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?
Answers
Answer:
Answer:
QS=2\sqrt{5}\text{ cm}QS=2
5
cm
RS=5\text{ cm}RS=5 cm
QR=3\sqrt{5}\text{ cm}QR=3
5
cm
Step-by-step explanation:
Give: PQR is a right angled triangle at Q and QS is perpendicular to PR. Please see the attachment for figure. If PQ=6cm and PS=4cm
To determine: QS, RS and QR
Calculation:
Using pythagoreous theorem, In ΔPQS, ∠S=90°
PQ^2=PS^2+QS^2PQ
2
=PS
2
+QS
2
6^2=4^2+QS^26
2
=4
2
+QS
2
QS=\sqrt{36-16}=2\sqrt{5}\text{ cm}QS=
36−16
=2
5
cm
In ΔPQS, ∠S=90°
\tan\theta =\frac{2\sqrt{5}}{4}tanθ=
4
2
5
In ΔPQR, ∠Q=90°
\tan\theta =\frac{QR}{6}tanθ=
6
QR
\frac{QR}{6}=\frac{2\sqrt{5}}{4}
6
QR
=
4
2
5
QR=3\sqrt{5}\text{ cm}QR=3
5
cm
Using pythagoreous theorem, In ΔRQS, ∠S=90°
RQ^2=RS^2+QS^2RQ
2
=RS
2
+QS
2
(3\sqrt{5})^2=RS^2+(2sqrt{5})^2(3
5
)
2
=RS
2
+(2sqrt5)
2
RS=\sqrt{45-20}=5\text{ cm}RS=
45−20
=5 cm
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
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)
Since 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)
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