Tangent Theorem

Q.

PQ is a tangent drawn from a point P to a circle with center O and QOR is a diameter of the circle such that $\angle$POR $={120}^{\circ }$, then $\angle$OPQ is ${30}^{\circ }$.

1. 60°

2. 45°

Q. prove that a rectangle that circumscribed a circle is a square.

Q.

ABC is an isosceles triangle in which AB = AC, circumscribed about a circle. Show that BC is bisected at the point of contact. [1 MARK]

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Q. Question 5
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is
(A) 4 cm
(B) 5 cm
(C) 6 cm
(D) 8 cm

Q.

In the given figure, AB is a tangent to the circle with centre O. If OP = PC, then $\angle OCP$ = ___.

1. ${30}^{\circ }$

2. ${45}^{\circ }$

3. ${60}^{\circ }$

4. ${15}^{\circ }$

Q. Two equal circles touch each other externally at C and AB is a common tangent to the circles . Then, ∠ACB =

(a) 60°

(b) 45°

(c) 30°

(d) 90°

Q.

A tangent PT is drawn parallel to a chord AB as shown in figure. Prove that APB is an isosceles triangle.

Q.

In the given figure, length of BC is:

1. 13 cm

2. 15 cm

3. 21 cm

4. 10 cm

Q.

State true or false:
For a tangent, the perpendicular line from the point of contact to the circle passes through the centre.

1. True

2. False

Q.

AP is the tangent to the circle with centre O and radius 8 cm . If AB = 9 cm, then find the length of tangent AP.

1. 10 cm

2. 8 cm

3. 15 cm

4. 13 cm

Q.

In the given figure ABCD is quadrilateral circumscribing a circle with centre O. If $\angle$B = ${90}^0$, AD = 23 cm, AB = 29 cm and DS = 5 cm, then the radius of the circle is ___ cm.

Q.

In the given figure, AB is a tangent to the circle with centre O. If OP = PC, then $\angle OCP$ = ___.

1. ${30}^{\circ }$

2. ${45}^{\circ }$

3. ${60}^{\circ }$

4. ${15}^{\circ }$

Q.

A line segment AB intersects a circle at two distinct points C and D as it passes through its centre, as shown in the figure. The line, whose segment is AB, is a:

1. Diameter

2. Chord

3. Tangent

4. Secant

Q. Three tangents are drawn at random to a given circle . Find the probability of the tangents to form a triangle .

Q.

Perimter of triangle ABC is 20 cm and AC = BC= 7 cm. CP = ____

1. 2 cm

2. 7 cm

3. 4 cm

4. 9 cm

Q. 24. If θ be the angle between two tangents which are drawn to the circle from the origin, then the value of cotθ is ?

Q.

In a circle, O is the centre and $\angle$COD is right angle. AC = BD and CD is the tangent at P. Which of the following are true, if the radius of the circle is 1 m?

1. BD = 41.42 cm

2. OD = 141.42 cm

3. PD = 100 cm

4. AC + CP = 141.42 cm

Q. In the following figure, AB is a chord of the circle and AOC is its diameter such that ∠ACB = 50°. If AT is the tangent to the circle at the point A, then ∠BAT is equal to

1. 65°
2. 60°
3. 50°
4. 40°

Q. Question 2 (iv)
Fill in the blanks:
(iv) The common point of a tangent to a circle and the circle is called___.

Q.

In the given circle, O is the centre and AD, AE are the two tangents. If BC is also a tangent, then .

1. 3AE = AB + BC + AC
2. AB + BC + AC = 4AE
3. 2AE = AB + BC + AC

Q.

In the given circle, O is the centre and AD, AE are the two tangents. If BC is also a tangent, then .

1. 3AE = AB + BC + AC
2. AB + BC + AC = 4AE
3. 2AE = AB + BC + AC

Q.

Prove that a parallelogram circumscribing a circle is a rhombus [4 MARKS]

Q. The given figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T.
Prove that ∠PTQ=2∠OPQ.

Q. Two parallel lines touch a circle at points A and B respectively. The area of the circle is $25\pi c{m}^2$ then AB is:

1. 10 cm
2. 25 cm
3. 5 cm
4. 8 cm

Q.

There are 2 questions titled A and B and there are 6 answers numbered from 1 to 6. In each of the answer options, each question is matched with an answer. Pick the correct match from the answer options.

A - What is the shortest distance between center of a circle and tangent?

B - What is the shortest distance between a point and a line?

1. Radius
2. Secant
3. Perpendicular
4. Chord
5. Arc
6. Tangent

1. A - 1

2. B - 3

3. A - 4

4. B - 2

Q.

In the given circle, O is the centre of the circle and AD, AE are the two tangents. If BC is also a tangent, then which of the given options is true?

1. AC + AB = BC

2. 3AE = AB + BC + AC

3. 2AE = AB + BC + AC

4. AB + BC + AC = 4AE

Q. From an external point P, two tangents are drawn that touch the circle at points Q and R. The centre of the circle is O, points O and P are joined. The ratio of ∠QPR and ∠OPQ is ______.

1. 2 : 1
2. 1 : 1
3. 4 : 1
4. 1 : 2

Q. In the given figure, O is the centre of the circle and AB is a tangent to it at point B. If CD and OA intersect at E on the circle and $\angle BDC={65}^{\circ }$, then find the measure of $\angle BAO$.

Q. In figure, AB is a chord of the circle and AOC is its diameter such that $\angle ACB={50}^{\circ }$. If AT is the tangent to the circle at the point A, then $\angle$ BAT is equal to

Q.

How can I tell if a line is tangent to a circle?