Circles

### Fill in the blanks. (i) A line intersecting a circle in two distinct points is called a ……… (ii) A circle can have parallel tangents at the most … (iii) The common point of a tangent to a circle and the circle is called the ……… (iv) A circle can have ….. tangents

Sol: A line intersecting a circle at two district points is called a secant A circle can have two parallel tangents at the most The common point of a tangent to a circle and the circle is called the...

### Which of the following statement is not true? (a) A line which intersect a circle in tow points, is called secant of the circle. (b)A line intersecting a circle at one point only, is called a tangent to the circle. (c) The point at which a line touches the circle, is called the point of contact. (d) A tangent to the circle can be drawn form a point inside the circle.

Answer: (d) A tangent to the circle can be drawn form a point inside the circle. Sol: A tangent to the circle can be drawn from a point Inside the circle. This statement is false because tangents...

### Which of the following statements is not true? (a) A tangent to a circle intersects the circle exactly at one point. (b) The point common to the circle and its tangent is called the point of contact. (c) The tangent at any point of a circle is perpendicular to the radius of the circle through the point of contact. (d) A straight line can meet a circle at one point only.

Answer: (d) A straight line can meet a circle at one point only. Sol: A straight be can meet a circle at one point only This statement is not true because a straight line that is not a tangent but a...

### Which of the following statements in not true? (a)If a point P lies inside a circle, not tangent can be drawn to the circle, passing through p. (b) If a point P lies on the circle, then one and only one tangent can be drawn to the circle at P. (c) If a point P lies outside the circle, then only two tangents can be drawn to the circle form P. (d) A circle can have more than two parallel tangents. parallel to a given line.

Answer: (d) A circle can have more than two parallel tangents. parallel to a given line. Sol: A circle can have more than two parallel tangents. parallel to a given line. This statement is false...

### Which of the following pairs of lines in a circle cannot be parallel? (a) two chords (b) a chord and tangent (c) two tangents (d) two diameters

Answer: (d) two diameters Sol: Two diameters cannot be parallel as they perpendicularly bisect each other.

### In a circle of radius 7 cm, tangent PT is drawn from a point P such that PT =24 cm. If O is the centre of the circle, then length OP = ?

(a) 30 cm                     (b) 28 cm                    (c) 25 cm                     (d) 18 cm Answer: (c) 25 cm Sol: The tangent at any point of a circle is perpendicular to the radius at the...

### In the given figure, a cradle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14cm, BC = 8cm and CA=12 cm. Find the length AD, BE and CF.

BD = 5cm = BE Solving (3) and (4), we get and AD = 9cm

### In the adjoining figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9cm and CD=8 cm. Find the length of side AD.

Sol: We know that when a quadrilateral circumscribes a circle then sum of opposites sides is equal to the sum of other opposite sides. \ AB + CD = AD + BC Þ 6 + 8 = AD = 9 Þ AD = 5 cm...

### In Fig below, PQ is tangent at point R of the circle with center O. If ∠TRQ = 30°, find ∠PRS

Given, \$\angle TRQ={{30}^{\circ }}\$ . At point R, OR ⊥ RQ. So, \$\angle ORQ={{90}^{\circ }}\$ \$\Rightarrow \angle TRQ+\angle ORT={{90}^{\circ }}\$ \$\Rightarrow \angle ORT={{90}^{\circ...

### If AB, AC, PQ are the tangents in the figure, and AB = 5 cm, find the perimeter of ∆APQ

Since AB and AC are the tangents from the same point A ∴AB=AC=5cm Similarly, BP=PX and XQ=QC Perimeter of \[\Delta APQ=AP+AQ+PQ\] \[=AP+AQ+(PX+XQ)\] \[=(AP+PX)+(AQ+XQ)\] \[=(AP+BP)+(AQ+QC)\]...

### A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.

Provided in question: Chord PQ is parallel to tangent at R.To prove: R bisects the arc PRQ. Proof: Since PQ || tangent at R. \$\angle 1=\angle 2\$  [alternate interior angles]\$\angle 1=\angle 3\$...

### Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

Suppose C1 and C2 are two circles with the same center O. And  AC is a chord touching C1 at the point D let’s join OD.So, \$OD\bot AC\$\$AD=DC=4cm\$  [perpendicular line OD...

### If the quadrilateral sides touch the circle, prove that sum of pair of opposite sides is equal to the sum of other pair.

Let’s Consider a quadrilateral ABCD touching circle with the centre O at points E, F, G and H as we can see in figure. We know that, In a circle with two points outside of it, the tangents drawn...

### If PT is a tangent at T to a circle whose centre is O and OP = 17 cm, OT = 8 cm. Find the length of the tangent segment PT.

Given in the question, OT = radius = \$8cm\$ OP = \$17cm\$ It is given to find: PT = length of tangent =\$?\$ T is point of contact. We also know that the tangent and radius are perpendicular at the point...

### Fill in the blanks:

(i) A circle can have …………… parallel tangents at the most. (ii) The common point of a tangent to a circle and the circle is called ………… Answer: (i) A circle can have two parallel tangents...

### Fill in the blanks:

(i) A tangent to a circle intersects it in …………… point(s). (ii) A line intersecting a circle in two points is called a …………. Answer: (i) A tangent to a circle intersects it...

### Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Answer: First, draw a quadrilateral ABCD that circumscribes a circle with centre O, touching the circle at points P, Q, R, and S. We now have the following figure after joining...

### A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Answer: The given figure: Considering the △ABC, We know that any two tangents drawn from the same point to the circle have the same length. As a result, (i) BE = BD = 8 cm (ii) CF = CD = 6 cm (iii)...

### Prove that the parallelogram circumscribing a circle is a rhombus.

Answer: Consider the parallelogram ABCD, which circumscribes a circle with O as centre. As ABCD is a parallelogram, so AB = CD and BC = AD. As seen in the figure above, (i) BP = BQ (ii) DR...

### Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

Answer: First, draw a circle with the centre O. Draw two tangents PA and PB at point A and point B, respectively, from an exterior point P. Now join A and B to form AB in such a way that it subtends...

### In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

Answer: From the given figure in the textbook, join OC. The diagram will now be as- Now using the SSS congruency the triangles △OPA and △OCA are similar: (i) OP = OC since they are the same circle’s...

### A quadrilateral ABCD is drawn to circumscribe a circle (see Fig. 10.12). Prove that AB + CD = AD + BC

Answer: The given figure is: We can draw a few conclusions from this figure, which are as follows: (i) BP = BQ (ii) DR = DS (iii) CR = CQ (iv) AP = AS Since the above drawn conclusions are tangents...

### Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Answer: With the centre O, draw two concentric circles. Now, in the larger circle, draw a chord AB that touches the smaller circle at a point P, as shown in the diagram below. AB is tangent to the...

### The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.

Answer: Draw a diagram according to the question. Here, AB is a tangent drawn on the circle from a point A. As a result, OB will be perpendicular to AB i.e. OB ⊥ AB We know that, AB =...

### Prove that the perpendicular at the point of contact to the tangent to a circle passes through the center.

Solution: Draw a circle with a centre O and a tangent AB that touches the circle's radius at point P. To Prove: PQ passes through point O. Consider the case where PQ does not pass through point...

### Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Answer: Firstly, draw a circle and connect two points A and B such that AB becomes the circle's diameter. Now, at points A and B, draw two tangents PQ and RS, respectively. Both radii, AO and...

### If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to

(A) 50° (B) 60° (C) 70° (D) 80° Answer: First, construct a diagram according to the statement given. Now, in the diagram above, OA represents the radius to tangent PA, while OB represents the radius...

### In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to

(A) 60° (B) 70° (C) 80° (D) 90° Answer: The radius of the circle to the tangent PT is OP, and the radius to the tangents TQ is OQ, as stated in the question. As a result, OP ⊥ PT and TQ ⊥ OQ...

### From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is

(A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm Answer: First draw a perpendicular from the triangle's centre O to a point P on the circle that touches the tangent. This line will be perpendicular to the...

### Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Answer: XY and AB are two parallel lines in the figure above. The line segment AB is the tangent at point C, while the secant is line segment XY.