Content Curator
Circle is the path traced by a moving point that moves in such a way that its distance from a fixed point remains the same. The fixed point is called the center of the circle and the fixed distance is called the radius of the circle. The distance from the center of the circle to the circumference always remains the same which means the radius of the circle is always of the same value no matter from what point from the circumference you measure.
| Table of Content |
Keyterms: Circle, radius, Line segment, Tangent, Secant, Right triangle
Secant and Tangents
[Click Here for Sample Questions]
When a line segment touches a circle it can touch it at one point or pass through it. When a line segment touches a circle at two points then it is called a secant. When a line segment touches a circle at one point then it is called a tangent. A tangent is a special case of secant .
Secant and Tangent
In the given figure Line Segment UY touches the circle at X. When the secant MN has moved away as M’N’ and M’’N’’ finally comes to a point M’’’N’’’ where it touches the circle O at only one point, it is called the tangent of the circle. That is why it is said that Tangent is a special Case of Secant.
Also Read:
| Related Articles | ||
|---|---|---|
| Tangent to a Circle | Slope of Secant Line | Central Angle of Circle Formula |
| Great Circle Formula | Unit Circle Formula | Areas Related to Circle |
Theorem Related To Tangent
[Click Here for Sample Questions]
Theorem 1: The tangent at any point of a circle and the radius through the point are perpendicular to each other.
To Prove: OP perpendicular to XY.
Proof: In figure 1.2 a circle with center O and tangent XY with point P at the interaction id given.
Take a point Q on XY other than P and join OQ.
The point Q must lie outside the circle. Therefore OQ is longer than the radius OP of the circle.
OP<OQ.
Since that happens for all the points on XY except point P, OP is the shortest of all the distances from point O to XY.
Thus, OP is perpendicular to XY.
Figure 1.2
In the given figure tangent T is perpendicular to the radius CT and at any given tangent, it is perpendicular to the radius of the circle.
Direct Common Tangent & Transverse Common Tangent
[Click Here for Sample Questions]
A common tangent is called a direct common tangent if both the circles lie on the same side of it whereas a common tangent is called a transverse common tangent if the circle lies on the opposite side of it.
Theorem 2: The length of tangents drawn from a common point outside the circle is equal.
Figure 1.3
To Prove: PA=PB
Proof: Consider a circle with center O as in figure 1.3.
A point P lying outside the circle with and two tangents PA, PB are drawn.
First join OP, OA, OB
Angles OAP and OBP are right angles because those are angles between radii and tangents and according to theorem 1, they are right angles.
Now in the right triangle OAP and OBP, OA=OB, ∠OAP =∠OBP.
Thus, PA =PB.
Hence proved.
Things to Remember
- Circle is a closed continuous curve with a fixed distance from a fixed point.
- Diameter is the biggest Chord of a circle.
- Perpendicular bisector of a chord passes through the center of the circle.
- Tangent is a special case of secant.
- Radius has the same distance from all the points on the circle.
- No tangent can be drawn from a point inside the circle.
- Two tangents can be drawn from a point outside the circle.
Also Read:
Sample Questions
Ques: The radius of a circle is 10 cm. Find the length of the chord which is at a distance of 6 cm from the center of the circle. (2 marks)
Ans: Radius of circle = 10 cm
length of perpendicular = 6cm
length of half of the chord =√10 2-√6 2 =√82 =8
Length of chord =8×2=16cm
Ques: If a point P is 17 cm from the center of a circle of radius 8 cm, then find the length of the tangent drawn to the circle from point P. (2 marks)
Ans:
OA ⊥ PA (√ radius is ⊥ to tangent at point of contact) ∴ In √OAP, we havePO2 = PA2 + AO2 ⇒ (17)2 = (PA)2 + (8)2 , (PA)2 = 289 – 64 = 225 ⇒ PA = √225 = 15
Hence, the length of the tangent from point P is 15 cm.
Ques: The length of the tangent to a circle from a point P, which is 25 cm away from the center, is 24 cm. What is the radius of the circle? (2 marks)
Ans:
√ OQ ⊥ PQ∴ PQ2 + QO2 = OP2 ⇒ 252 = OQ2 + 242
or
OQ = √625 – √576 = √49 = 7 cm.
Ques:PQ is the chord of a given circle on producing it meets a tangent TR at R.If PQ=5 cm and QR=4 cm, Find TR. (2 marks)
Ans: PR =PQ+QR = 9cm
PR ⊥ QR=TR2
=9 ⊥ 4= 36
TR = 6 cm
Ques: Draw two concentric circles with their centres at O. OP = 4cm and OQ = 5cm. AB is the chord of the outer circle and a tangent to the inner circle at P. Find the length of AB. (2 marks)
Ans: Radius OP is perpendicular to tangent AB.
Therefore, OP bisects AB as perpendicular from center bisects a chord.
So, AB =2PB,OB=OQ=5cm
In triangle OPB, PB2 =OB2-OP2 = 52 -4 = 9cm2.
PB=3cm
Thus AB=6cm.
Ques: Prove that the tangents drawn at the ends of a diameter of a circle are parallel. (5 marks)
Ans: First, draw a circle and connect two points A and B such that AB becomes the diameter of the circle. Now, draw two tangents PQ and RS at points A and B. Now, both radii i.e. AO and OB are perpendicular to the tangents.
So, OB is perpendicular to RS and OA perpendicular to PQ
So, ∠OAP = ∠OAQ = ∠OBR = ∠OBS = 90°
From the above figure, angles OBR and OAQ are alternate interior angles.
Also, ∠OBR = ∠OAQ and ∠OBS = ∠OAP {since they are also alternate interior angles}
So, it can be said that line PQ and the line RS are parallel to each other.
Ques: How many tangents can be drawn from the external point to a circle? (2 marks)
Ans: Two tangents can be drawn from the external point to a circle.
Ques: Prove that the lengths of tangents drawn from an external point to a circle are equal. (5 marks)
Ans: Consider a circle with the center “O” and P is the point that lies outside the circle.
Hence, the two tangents formed are PQ and PR.
We need to prove: PQ = PR.
To prove the tangent PQ is equal to PR, join OP, OQ and OR.
Hence, ∠OQP and ∠ORP are the right angles
Therefore, OQ = OR (Radii)
OP = OP (Common side) ,
By using the RHS rule, we can say, ? OQP ≅ ? ORP.
Thus, by using the CPCT rule, the tangent PQ = PR.
Ques: A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is (2 marks)
(A) 12 cm. (B) 13 cm (C) 8.5 cm (D) √119 cm
Ans: We know that the line drawn from the center of the circle to the tangent is perpendicular to the tangent.
OP perpendicular PQ
By applying Pythagoras theorem in ΔOPQ,
OP² + PQ² = OQ²
5² + PQ² =122
PQ² =144 − 25
PQ = √119 cm.
Hence, the correct answer is (D).
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Also Check:
Comments