Advertisements
Advertisements
Question
Length of a tangent segment drawn from a point which is at a distance 15 cm from the centre of a circle is 12 cm, find the diameter of the circle?
Advertisements
Solution
Let seg BC be the tangent to the circle with centre A from point C.
∴ AC = 15 cm, BC = 12 cm ...[Given]
AB is the radius of the circle.
∴ In ∆ABC, ∠ABC = 90° ...[Tangent theorem]
∴ AC2 = AB2 + BC2
∴ (15)2 = AB2 + (12)2
∴ 225 = AB2 + 144
∴ AB2 = 225 – 144
∴ AB2 = 81
∴ AB = 9 ...[Taking square root of both sides]
i.e., radius of the circle is 9 cm.
∴ The diameter of the circle is 18 cm.
APPEARS IN
RELATED QUESTIONS
In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
- What is the length of each tangent segment?
- What is the measure of ∠MRO?
- What is the measure of ∠MRN?
Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity.
In the given figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii of the circles are 4 cm, 6 cm. 
In the given figure, seg EF is a diameter and seg DF is a tangent segment. The radius of the circle is r. Prove that, DE × GE = 4r2
Four alternative answers for the following question is given. Choose the correct alternative.
Length of a tangent segment drawn from a point which is at a distance 12.5 cm from the centre of a circle is 12 cm, find the diameter of the circle.
In the given figure, M is the centre of the circle and seg KL is a tangent segment.
If MK = 12, KL = \[6\sqrt{3}\] then find –
(1) Radius of the circle.
(2) Measures of ∠K and ∠M.
In the given figure, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle is r and `l`(AB) = r, Prove that ▢ABOC is a square. 
Proof: Draw segment OB and OC.
`l`(AB) = r ...[Given] (i)
AB = AC ...[`square`] (ii)
But OB = OC = r ...[`square`] (iii)
From (i), (ii) and (iii)
AB = `square` = OB = OC = r
∴ Quadrilateral ABOC is `square`
Similarly, ∠OBA = `square` ...[Tangent Theorem]
If one angle of `square` is right angle, then it is a square.
∴ Quadrilateral ABOC is a square.
In the following figure ‘O’ is the centre of the circle.
∠AOB = 1100, m(arc AC) = 450.
Use the information and fill in the boxes with proper numbers.
(i) m(arcAXB) =
(ii)m(arcCAB) =
(iv)∠COB =
(iv)m(arcAYB) =
The perpendicular height of a cone is 12 cm and its slant height is 13 cm. Find the radius of the base of the cone.
Prove the following theorem:
Tangent segments drawn from an external point to the circle are congruent.
The chords corresponding to congruent arcs of a circle are congruent. Prove the theorem by completing following activity.
Given: In a circle with centre B
arc APC ≅ arc DQE
To Prove: Chord AC ≅ chord DE
Proof: In ΔABC and ΔDBE,
side AB ≅ side DB ...`square`
side BC ≅ side `square` ...`square`
∠ABC ≅ ∠DBE ...[Measure of congruent arcs]
∆ABC ≅ ∆DBE ...`square`
Tangent segments drawn from an external point to a circle are congruent, prove this theorem. Complete the following activity.
Given: `square`
To Prove: `square`
Proof: Draw radius AP and radius AQ and complete the following proof of the theorem.
In ∆PAD and ∆QAD,
seg PA ≅ `square` ...[Radii of the same circle]
seg AD ≅ seg AD ...[`square`]
∠APD ≅ ∠AQD = 90° ...[Tangent theorem]
∴ ∆PAD ≅ ∆QAD ...[`square`]
∴ seg DP ≅ seg DQ ...[`square`]
In the adjoining figure, O is the center of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
- What is the length of each tangent segment?
- What is the measure of ∠MRO?
- What is the measure of ∠MRN?
In the adjoining figure circle with Centre, Q touches the sides of ∠MPN at M and N. If ∠MPN = 40°, find measure of ∠MQN.
If AB and CD are the common tangents in the circles of two unequal (different) radii, then show that seg AB ≅ seg CD.
Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON with the help of activity.
Proof: In ∆RMO and ∆RNO,
∠RMO ≅ ∠RNO = 90° ...[`square`]
hypt OR ≅ hypt OR ...[`square`]
seg OM ≅ seg `square` ...[Radii of the same circle]
∴ ∆RMO ≅ ∆RNO ...[`square`]
∠MOR ≅ ∠NOR
Similairy ∠MRO ≅ `square` ...[`square`]
