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Byju's Answer
Standard XII
Mathematics
Equation of Circle with (h,k) as Center
In the given ...
Question
In the given figure two tangents
P
Q
and
P
R
are drawn to a circle with centre
O
from an external point
P
. Prove that
∠
Q
P
R
=
2
∠
O
Q
R
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Solution
Given that PQ and QR are two tangents drawn to a circle with centre O from an external point P.
To prove:
∠
Q
P
R
=
2
∠
O
Q
R
Construction: Join QR, OQ and OR.
Proof: We know that lengths of a tangent drawn from an external point to a circle are equal.
P
Q
=
Q
R
Δ
P
Q
R
is an isosceles triangle
∠
P
Q
R
=
∠
P
R
Q
In
Δ
P
Q
R
∠
P
Q
R
+
∠
P
R
Q
+
∠
Q
P
R
=
180
o
∠
P
Q
R
+
∠
P
Q
R
+
∠
Q
P
R
=
180
o
2.
∠
P
Q
R
=
180
o
−
∠
Q
P
R
∠
P
Q
R
=
1
2
(
180
o
−
∠
Q
P
R
)
∠
P
Q
R
=
90
o
−
1
2
∠
Q
P
R
1
2
∠
Q
P
R
=
90
o
−
∠
P
Q
R
…………
(
1
)
Since PQ is perpendicular to PQ.
∠
O
Q
P
=
90
o
∠
O
Q
R
+
∠
P
Q
R
=
90
o
∠
O
Q
R
=
90
−
∠
P
Q
R
………..
(
2
)
⇒
∠
O
Q
R
=
1
2
⋅
∠
O
Q
R
⇒
2.
∠
O
Q
R
=
∠
Q
P
R
∴
∠
Q
P
R
=
2
∠
O
Q
R
Hence, it proved.
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Similar questions
Q.
In Fig. 3, two tangents PQ are PR are drawn to a circle with centre O from an external point P. Prove that ∠QPR = 2 ∠OQR.