English

In the following figure, XY = 10 cm and LT = 4 cm. Find the length of XT. - Geometry Mathematics 2

Advertisements
Advertisements

Question

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

Sum

Solution

Given: XY = 10 cm and LT = 4 cm

Congruent tangent segments are those drawn from an exterior point to a circle.

Thus, XY = XZ = 10 cm

Similarly, LT = TZ = 4 cm

Now, XT = XZ – TZ

= 10 – 4

= 6 cm

Hence, the length of XT is 6 cm.

shaalaa.com
Tangent Segment Theorem
  Is there an error in this question or solution?
2024-2025 (March) Model set 1 by shaalaa.com

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

  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, 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


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) =


In the given figure, M is the centre of the circle and seg KL is a tangent segment. L is a point of contact. If MK = 12, KL = `6sqrt3`, then find the radius of the circle.


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

(i) What is the length of each tangent segment?

(ii) What is the measure of ∠MRO?

(iii) 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 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`]


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



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)


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×