The Length of Two Tangents Drawn from an Exterior Point to a Circle Are Equal

Q.

Every rhombus is a rectangle. State with reasons whether the given statement is ‘true’ or ‘false’.

Q.

Two tangents PA and PB are drawn to the circle with centre O, such that $\angle$APB $={120}^{\circ }$. What is the relation between OP and AP?

1. $OP=\frac{1}{2}AP$

2. $OP=2AP$

3. $AP=\frac{2}{3}OP$

4. $OP=AP$

Q. Given figure shows a triangle ABC circumscribing a circle. If $\angle BOC={110}^{\circ }$, then $\angle BAC=$

1. ${40}^{\circ }$
2. ${50}^{\circ }$
3. ${35}^{\circ }$

Q.

Calculate the angle $X$ and angle $Y$ value in the given figure?

Q. Given figure shows, a right $△$ circumscribing a circle. If AP = 8 cm and BQ = 4 cm, then the length of AC =

1. 16 cm
2. 20 cm
3. 24 cm

Q. In the given figure, PA and PB are two tangents drawn from an external point P. If $\angle APB={40}^{\circ }$, then $\angle AOB=$

1. ${110}^{\circ }$
2. ${140}^{\circ }$
3. ${290}^{\circ }$

Q. In the given figure, A rigtht angled $△$ABC at B is circumscribing a circle. If AP = 8 cm and BQ = 4 cm, then the length of AC = .

1. 24 cm
2. 16 cm
3. 20 cm

Q. In the given figure, two concentric circles are drawn. If the radius of an outer circle = 13 cm and the length of the chord of an outer circle which touches an inner circle = 24 cm then, find the radius of inner circle.

1. 6 cm
2. 5.5 cm
3. 5 cm
4. 4 cm

Q. A pair of perpendicular tangents are drawn to a circle from an external point. Prove that length of each tangent is equal to the radius of the circle.

Q. From an external point P, two tangents are drawn that touches the circle at points Q and R. Then triangle PQR is a/an triangle.

1. scalene
2. right-angled
3. equilateral
4. isosceles

Q.

You are given a circle with radius ‘r’ and centre O. You are asked to draw a pair of tangents which are inclined at an angle of ${60}^{\circ }$ with each other. Refer to the figure and select the option which would lead us to the required construction. 'd' is the distance OE.

1. Construct the $\Delta MNO$ as it is equilateral

2. Mark M and N on the circle such that $\angle MOE={60}^{\circ }$ and $\angle NOE={60}^{\circ }$

3. Using trigonometry, arrive at $d=r\left(\sqrt{3}\right)$ and mark E.

4. None of the above

Q.

All the sides of the triangle ABC are tangential to the circle with centre R. What is the perimeter of $\Delta ABC$?

1. 28 cm

2. 20 cm

3. 5 cm

4. 30 cm

Q.
In the figure given above, PA and PB are tangents to the circle from point P. How many real values of x exist such that the length of PA is x and length of PB is $x^2+1$?

1. 0
2. 1
3. 2
4. 3

Q.

For the given quadrilateral, find BC + AD.

1. 14 cm

2. 20 cm

3. 19 cm

4. 15 cm

Q.

Two tangents are drawn from a point P to a circle touching the circle at A and B respectively . If PA = 6 cm and AB = 3 cm, then find the perimeter of triangle ABP .

1. 15 cm

2. 30 cm

3. 36 cm

4. 12 cm

Q. Match the following:

1. AP + DR = 5 cm
2. DC = 7 cm
3. DC = 9 cm

Q. $\text{In the given figure, ABCD is a}\text{quadrilateral circumscribing a circle with}\text{centre 'O'. If}\angle B={90}^{\circ },AD=23cm,AB=29cm\text{and}DS=5cm,\text{then find}\text{the radius of the circle.}$

1. 11 cm
2. 12 cm
3. 14 cm
4. 9 cm

Q. A tangent from a point P outside the circle meets the circle at M. Then $\angle MOP=$ , if $\angle MPO={30}^{\circ }$.

1. ${30}^{\circ }$
2. ${60}^{\circ }$
3. ${45}^{\circ }$

Q.

In the figure; PA is a tangent to the circle. PBC is secant and AD bisects angle BAC. Select the statements that are true.

1. $\angle CAD=\frac{1}{3}(\angle PBA-\angle PAB)$

2. Triangle PAD is an isoscles triangle

3. $\angle CAD=\frac{1}{2}(\angle PBA-\angle PAB)$

4. $\angle CAD=\frac{1}{9}(\angle PBA-\angle PAB)$

Q.

Which of the following statements fits best as per the diagram given below.

1. PA = PD

2. PA = AQ

3. BQ = DS

4. RC = DS