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महाराष्ट्र राज्य शिक्षण मंडळएस.एस.सी (इंग्रजी माध्यम) इयत्ता १० वी

Segment DP and segment DQ are tangent segments to the circle with center A. If DP = 7 cm. So find the length of the segment DQ?

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प्रश्न

Segment DP and segment DQ are tangent segments to the circle with center A. If DP = 7 cm. So find the length of the segment DQ?

बेरीज
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उत्तर

Seg DP and seg DQ are the tangents to the circle.

∴ DP = DQ   ...[Tangent segment theorem]

∴ DQ = 7 cm   ...[Given]

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Tangent Segment Theorem
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पाठ 3: Circle - Exercise

संबंधित प्रश्‍न

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

  1. What is the length of each tangent segment?
  2. What is the measure of ∠MRO?
  3. What is the measure of ∠MRN?


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, 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.


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.


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`


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

  1. What is the length of each tangent segment?
  2. What is the measure of ∠MRO?
  3. What is the measure of ∠MRN?


The figure ΔABC is an isosceles triangle with a perimeter of 44 cm. The sides AB and BC are congruent and the length of the base AC is 12 cm. If a circle touches all three sides as shown in the figure, then find the length of the tangent segment drawn to the circle from the point B.


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`]


Prove that, tangent segments drawn from an external point to the circle are congruent.


In a parallelogram ABCD, ∠B = 105°. Determine the measure of ∠A and ∠D.


In the following figure, XY = 10 cm and LT = 4 cm. Find the length of XT.


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