This MCQ module is based on: Tangent to a Circle
TOPIC 25 OF 35
Tangent to a Circle
🎓 Class 10
Mathematics
CBSE
Theory
Ch 10 — Circles
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Tangent to a Circle
Targeting Class 10 level in Geometry, with Intermediate difficulty.
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10.1 Introduction
In Class IX you learned that a circle is the collection of all points in a plane that are at a constant distance, called the radius, from a fixed centre. You also studied many terms associated with circles: chord, segment, sector, arc and so on. This chapter now focuses on a new viewpoint — when lines in the plane meet circles, how many crossings can there be, and what special properties does the kissing case enjoy? That case is called a tangenti.
Consider a circle and a line PQ in the same plane. Three cases can occur:
Fig. 10.1 — Three possibilities for a line and a circle in a plane
- (i) Non-intersecting line: the line and circle have no common point.
- (ii) Tangent: the line touches the circle at exactly one common point, called the point of contacti.
- (iii) Secant: the line cuts the circle at two distinct points.
You may have watched a bicycle in motion — a point on the wheel's rim touches the road for just an instant. The road behaves like a tangent to the wheel. Similarly, a taut cable that supports a stream of motion, a belt wrapped round a pulley, and a billiard ball grazing the cushion of the table all illustrate the idea of a tangent line.
Activity 1 — The rotating ruler
Take a thin circular card and a ruler. Hold the ruler as a secant so it cuts the circle at two points A and B. Rotate the ruler slowly about one of the points, say A. As you rotate, the chord AB shrinks. At a particular position, B coincides with A — the line then touches the circle at a single point. That line is the tangent at A, and the common point A is the point of contact.
A tangent to a circle is the limiting case of a secant when its two crossing points merge into one. This also suggests that at the point of contact, the tangent represents the direction in which a point on the secant is travelling.
10.2 Tangent to a Circle
Definition
A tangent to a circle is a line that intersects the circle at exactly one point. This unique common point is called the point of contact.Note two crucial facts right away:
- A tangent can be drawn at every point of a circle.
- At any given point of a circle, there is exactly one tangent (which we will prove via Theorem 10.1 below).
Activity 2 — Paper circle and a straight wire
Draw a circle of centre O and radius 6 cm. Insert a straight wire AB across the paper so that it forms a chord. Now slide the wire parallel to itself, moving away from O. At a critical moment the wire lies along the circumference touching at exactly one point. Beyond that, it separates from the circle entirely.
Observe: the distance from O to the wire, measured perpendicular to the wire, equals the radius at the very moment of tangency. This experimentally confirms Theorem 10.1 that follows.
The tangent at a point of a circle is perpendicular to the radius through that point. Any line whose perpendicular distance from the centre is greater than the radius lies outside the circle; less than the radius, it is a secant; equal to the radius, it is a tangent.
Theorem 10.1 — Tangent ⊥ Radius
Theorem 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.Fig. 10.2 — Circle with centre O, tangent XY at P, point Q on XY outside the circle
Proof. Given a circle with centre O and a tangent XY to the circle at a point P. We need to prove that OP ⊥ XY.
Pick any point Q on XY other than P. Join OQ. Because XY is a tangent, it touches the circle only at P, so Q lies outside the circle. Therefore OQ is longer than the radius OP:
\[OQ>OP.\]This inequality holds for every point Q on XY other than P. In other words, OP is the shortest distance from O to the line XY. But we know from plane geometry that the shortest distance from a point to a line is the perpendicular segment.
Hence OP ⊥ XY, and the theorem is proved. \(\blacksquare\)
Remarks
Remark 1. By the theorem, we can now say conclusively that at any point on a circle there is one and only one tangent.
Remark 2. The line containing the radius through the point of contact is called the normal to the circle at that point. The tangent is perpendicular to this normal.
Exercise 10.1
Q1. How many tangents can a circle have?
Infinitely many — one tangent at each point on the circle, and the circle has infinitely many points.
Q2. Fill in the blanks: (i) A tangent to a circle intersects it in ____ point(s). (ii) A line intersecting a circle in two points is called a ____. (iii) A circle can have ____ parallel tangents at most. (iv) The common point of a tangent to a circle and the circle is called ____.
(i) one. (ii) secant. (iii) two (parallel tangents occur at diametrically opposite points). (iv) point of contact.
Q3. A tangent PQ at point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ = 12 cm. Length PQ is: (A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.
Since tangent is ⊥ radius at P, triangle OPQ is right-angled at P. PQ² = OQ² − OP² = 144 − 25 = 119. So PQ = √119 cm, option (D).
Q4. Draw a circle and two lines parallel to a given line such that one of them is a tangent and the other a secant to the circle.
Draw circle and given line ℓ. A line parallel to ℓ at perpendicular distance r (radius) from the centre is a tangent; any parallel at perpendicular distance less than r is a secant; any parallel at distance greater than r is non-intersecting.
Competency-Based Questions
Q1. A coin of radius 3 cm rolls along the inside of a larger track. At the instant the coin touches the track, apply Theorem 10.1 to describe the direction of the coin's centre-to-contact radius.
L3 ApplyThe track at the contact point acts as a tangent to the coin. By Theorem 10.1 the coin's radius to the point of contact is perpendicular to the track at that point — pointing into the coin's interior along the inward normal.
Q2. Analyse why a line drawn through an interior point of a circle can never be a tangent.
L4 AnalyseAny line through an interior point crosses the boundary at exactly two points (entering and exiting the disc). Hence it is always a secant, never a tangent.
Q3. Evaluate the claim: "If a line is perpendicular to a chord of a circle, it must be a tangent to the circle."
L5 EvaluateFalse. A line perpendicular to a chord need not meet the circle at only one point. The claim is true only when the perpendicular is drawn through the centre AND the chord is replaced by the diameter — i.e., the tangent at an endpoint of that diameter. In general, perpendicularity of a line to a chord is unrelated to being a tangent.
Q4. Design an experiment using a compass and straight-edge to verify Theorem 10.1 empirically.
L6 CreateProcedure: (1) Draw a circle of centre O. (2) Mark point P on the circle. (3) Draw a secant near P with two close crossing points A, B. (4) Measure angle between OP and line AB. (5) Slide AB toward the tangent position — measure the angle again. (6) Plot angle vs. chord length: as chord → 0, angle → 90°. This demonstrates that at tangency the radius-to-line angle becomes exactly 90°.
Assertion & Reason
A: A line that touches a circle at just one point is perpendicular to the radius at that point.
R: Among all segments from the centre to points on the tangent, the shortest is the perpendicular, which equals the radius.
R: Among all segments from the centre to points on the tangent, the shortest is the perpendicular, which equals the radius.
Answer: A. Precisely the argument of Theorem 10.1.
A: A circle has exactly two tangents parallel to any given line.
R: The perpendiculars from the centre to parallel tangents coincide as the diameter, giving exactly two such tangents at the diameter's endpoints.
R: The perpendiculars from the centre to parallel tangents coincide as the diameter, giving exactly two such tangents at the diameter's endpoints.
Answer: A. Any direction has exactly two tangents to a circle — one on each side of the centre.
A: If OP = 5 cm and the tangent length from an external point Q is 12 cm, then OQ = 13 cm.
R: Triangle OPQ is right-angled at P by Theorem 10.1, so OQ² = OP² + PQ².
R: Triangle OPQ is right-angled at P by Theorem 10.1, so OQ² = OP² + PQ².
Answer: A. √(25 + 144) = 13.
Frequently Asked Questions
What is the difference between a tangent and a secant?
A tangent touches a circle at exactly one point; a secant cuts the circle at exactly two points. A tangent can be thought of as a limiting position of a secant.
Why is the radius perpendicular to the tangent at the point of contact?
Because the perpendicular from the centre of a circle to any tangent is the shortest distance from the centre to that line, and this shortest distance equals the radius.
How many tangents can be drawn to a circle from an external point?
Exactly two tangents can be drawn from any external point to a circle, and they are of equal length.
How many tangents pass through a point on the circle?
Exactly one tangent passes through a given point on the circle, and it is perpendicular to the radius at that point.
Can a tangent pass through the centre of the circle?
No. The tangent touches the circle at just one point and the centre is inside the circle, so no tangent can pass through the centre.
What is Theorem 10.1 in Class 10 Circles?
Theorem 10.1 states: the tangent at any point of a circle is perpendicular to the radius through the point of contact.
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