Advertisements
Advertisements
प्रश्न
In the below given figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If∠PRQ = 120°, then prove that OR = PR + RQ.
उत्तर
Given that m∠PRQ = 120º
We know that the line joining the centre and the external point is the angle bisector between the tangents.
Thus, m∠PRO = m∠QRO =120º/2= 60º
Also we know that lengths of tangents from an external point are equal.
Thus, PR = RQ.
Join OP and OQ.
Since OP and OQ are the radii from the centre O,
OP ⊥ PR and OQ ⊥ RQ.
Thus, Δ OPR and ΔOQR are right angled congruent triangles.
Hence, ∠POR=90º- ∠PRO=90º - 60º= 30º
∠QOR=90°- ∠QRO=90° - 60° =30°
sin ∠QRO = sin30°=1/2
Thus,PR/OR=1/ 2
OR =2PR
OR =PR + PR
OR =PR + QR
APPEARS IN
संबंधित प्रश्न
If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.
Find the area of the shaded region in Fig. 8, where \\
M and N are the midpoints of chords AB and CD . The line MN passes through the centre O . Prove that AB || CD.
The number of tangents drawn at a point of the circle is/are ______
The length of the tangent from an external point on a circle is ______
PA and PB are the two tangents drawn to the circle. O is the centre of the circle. A and B are the points of contact of the tangents PA and PB with the circle. If ∠OPA = 35°, then ∠POB = ______
Draw two concentric circles of radii 2 cm and 5 cm. From a point P on outer circle, construct a pair of tangents to the inner circle.
In the given figure, PQ and PR are tangents drawn from P to the circle with centre O such that ∠QPR = 65°. The measure of ∠QOR is ______.
A quadrilateral PQRS is drawn to circumscribe a circle. If PQ = 12 cm, QR = 15 cm and RS = 14 cm, then find the length of SP is ______.
In the given diagram, O is the centre of the circle. PR and PT are two tangents drawn from the external point P and touching the circle at Q and S respectively. MN is a diameter of the circle. Given ∠PQM = 42° and ∠PSM = 25°.
Find:
- ∠OQM
- ∠QNS
- ∠QOS
- ∠QMS
