The tangents drawn at the extremities of the diameter of a circle are always:
[1]
In the given figure, PA is a tangent from an external point P to a circle with centre O. If $\angle POB=115^{\circ}$ then $\angle APO$ is equal to:
[1]
A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.
(a) In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that $\angle BAC+\angle ACD=90^{\circ}$.
OR
(b) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
[3]
Directions: In Q. No. 19 and 20 a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
(a) Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
(b) Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
Assertion (A): The tangents drawn at the end points of a diameter of a circle, are parallel.
Reason (R): Diameter of a circle is the longest chord.
[1 mark]
In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle. Prove that $\angle POQ=90^{\circ}$.
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[3 marks]
A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the points of contact as shown. If AD is perpendicular to DC, $BC=30$ cm and $BS=24$ cm, then find the length DC.
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[3 marks]
A backyard is in the shape of a triangle ABC with right angle at B. $AB=7~m$ and $BC=15$ m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that $AP=xm.$
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Based on the above information, answer the following questions:
(i) Find the length of AR in terms of x.
(ii) Write the type of quadrilateral BQOR.
(iii) (a) Find the length PC in terms of x and hence find the value of x.
OR
(b) Find x and hence find the radius r of circle.
[4 marks]
In the given figure, PQ is tangent to the circle centred at O. If \( \angle AOB=95^{\circ} \) then the measure of \( \angle ABQ \) will be
[1 mark]
(a) Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that \( \angle PTQ=2\angle OPQ \).
OR
(b) In the given figure, a circle is inscribed in a quadrilateral ABCD in which \( \angle B=90^{\circ} \). If \( AD=17 \text{ cm} \), \( AB=20 \text{ cm} \) and \( DS=3 \text{ cm} \), then find the radius of the circle.
[3 marks]
Case Study
The discus throw is an event in which an athlete attempts to throw a discus. The athlete spins anti-clockwise around one and a half times through a circle, then releases the throw. When released, the discus travels along tangent to the circular spin orbit.
In the given figure, AB is one such tangent to a circle of radius 75 cm. Point O is centre of the circle and \( \angle ABO=30^{\circ} \). PQ is parallel to OA.
Based on above information:
- (a) find the length of AB.
- (b) find the length of OB.
- (c) find the length of AP.
OR
(c) Find the length of PQ
[4 marks]
Two concentric circles are of radii 4 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
[2 marks]
In Figure 1, a triangle ABC with \(\angle B=90^{\circ}\) is shown. Taking AB as diameter, a circle has been drawn intersecting AC at point P. Prove that the tangent drawn at point P bisects BC.
[4 marks]
