Advertisements
Advertisements
Question
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?
Solution
(1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively.
∴ ∠OMR = ∠ONR = 90º ...(Tangent at any point of a circle is perpendicular to the radius throught the point of contact)
OM = 5 cm and OR = 10 cm
In right ∆OMR,
OR2 = OM2 + MR2
MR = `sqrt("OR"^2 - "OM"^2`
MR = `sqrt(10^2 - 5^2)`
MR = `sqrt(100 - 25)`
MR = `sqrt75`
MR = 5`sqrt3` cm
Tangent segments drawn from an external point to a circle are congruent.
∴ MR = NR = 5`sqrt3`
(2) In right ∆OMR,
∠MRO = `"OM"/"MR"`
∠MRO = `(5 "cm")/(5sqrt3cm)`
= `1/sqrt3`
∠MRO = tan 30°
∠MRO = 30°
Thus, the measure of ∠MRO is 30º.
Similarly, ∠NRO = 30º
(3) ∠MRN = ∠MRO + ∠NRO = 30º + 30º = 60º
Thus, the measure of ∠MRN is 60º.
RELATED QUESTIONS
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, O is the centre of the circle and B is a point of contact. seg OE ⊥ seg AD, AB = 12, AC = 8, find (1) AD (2) DC (3) DE.
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
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 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.
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`
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?
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
(i) What is the length of each tangent segment?
(ii) What is the measure of ∠MRO?
(iii) 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.
Prove that, tangent segments drawn from an external point to the circle are congruent.
In the following figure, XY = 10 cm and LT = 4 cm. Find the length of XT.
A circle touches side BC at point P of the ΔABC, from outside of the triangle. Further extended lines AC and AB are tangents to the circle at N and M respectively. Prove that : AM = `1/2` (Perimeter of ΔABC)
