MathsCBSE Class 10 · Circles

Tangent and Secant of a Circle for Class 10

Updated 14 October 2026 · Fact-checked

A secant is a line that cuts a circle at two points. A tangent touches it at exactly one point, called the point of contact. From a point inside a circle no tangent can be drawn, from a point on it exactly one, and from a point outside exactly two. The board tests these through MCQs, assertion-reason and short problems.

Understand Tangent and Secant of a Circle

Put a straight line and a circle on the same page. There are only three ways they can sit together. The line can miss the circle completely (a non-intersecting line). It can cut the circle at two points (a secant). Or it can just touch the circle at one point (a tangent). That single shared point is the point of contact, and the tangent is said to touch the circle there.

Why only these three? Drop a perpendicular from the centre $O$ to the line and call its length $d$ . Compare $d$ with the radius $r$ . If $d > r$ , every point of the line is further than $r$ from $O$ , so the line misses the circle. If $d < r$ , the foot of the perpendicular is inside the circle, and the line must cross the circle once on each side of it, giving two points. If $d = r$ , the foot is on the circle and every other point of the line is further away, so the line meets the circle only there. This is the distance test: $d > r$ misses, $d = r$ tangent, $d < r$ secant.

You can also picture a tangent as a secant pushed to its limit. Take a secant cutting the circle at $A$ and $B$ , and slide the line parallel to itself away from the centre. The chord $A B$ gets shorter, and when $A$ and $B$ merge into one point, the secant has become a tangent. For a secant at distance $d$ , the chord has length $2 r^{2} - d^{2}$ , which becomes $0$ exactly when $d = r$ . Notice that for any given direction, exactly two lines are tangent to the circle, one on each side of the centre.

Now the number of tangents from a point $P$ . If $P$ is inside the circle, every line through $P$ passes through an inside point, so it is a secant: no tangent. If $P$ is on the circle, there is exactly one tangent, the line through $P$ perpendicular to the radius $O P$ . If $P$ is outside, there are exactly two tangents, one on each side of line $O P$ . To find where $P$ is, compare $O P$ with $r$ , using the distance formula when coordinates are given.

These ideas set up the whole chapter. The perpendicular-to-radius theorem and the equal-tangents theorem, studied on the next pages, both assume you know what a tangent and a point of contact are, and that two tangents exist from an external point.

Tangent and Secant of a Circle: formulas to remember

Distance test for a line and a circle: $d > r : no common point, d = r : tangent, d < r : secant$; d is the perpendicular distance from the centre to the line, r is the radius.
Length of chord cut by a secant: $chord = 2 r^{2} - d^{2}$; The perpendicular from the centre bisects the chord. The chord shrinks to 0 when d = r, the tangent position.
Number of tangents from a point P: $O P < r : 0, O P = r : 1, O P > r : 2$; Use the distance formula $O P = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ when coordinates are given.
Tangents parallel to a given line: $exactly 2, each at distance r from O, gap between them = 2 r$; One on each side of the centre.
Total tangents of a circle: $one at each point \Rightarrow infinitely many$; A circle has infinitely many points, and exactly one tangent at each.

How to solve Tangent and Secant of a Circle questions

Every question on this topic comes down to comparing a distance from the centre with the radius. Use this routine.

1Identify the centre $O$ and the radius $r$ . If only a point on the circle is given, find $r$ with the distance formula first.
2Decide what you are comparing: for a line, its perpendicular distance $d$ from $O$ ; for a point $P$ , the distance $O P$ .
3Compute the distance cleanly. Compare squares ( $d^{2}$ with $r^{2}$ ) to avoid square roots.
4Classify using the rule: smaller than $r$ means inside or secant, equal means on or tangent, greater means outside or missing.
5For a line, state the number of common points (2, 1 or 0); for a point, state the number of tangents (0, 1 or 2) with the reason.
6If a chord length is asked, use the perpendicular from the centre bisecting the chord and write $chord = 2 r^{2} - d^{2}$ .
7If parallel lines are involved, remember the tangents in that direction sit at distance $r$ on both sides of $O$ , so check both cases.

Tangent and Secant of a Circle in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Tangent, secant and the distance test

Imagine your family draws a circular rangoli of radius 1.3 m in the courtyard for a festival. Your sister lays a straight chalk line for a row of diyas, and its perpendicular distance from the rangoli's centre is 1.2 m. She wants the diya line to just touch the rangoli instead.

How long is the part of the chalk line that lies inside the rangoli? By how much must the line be shifted, parallel to itself, so that it just touches the rangoli? Give both possible answers.

Try it first, then show the solutionSolution

Here $r = 1.3$ m and $d = 1.2$ m, and $d < r$ , so the line is a secant.
Half the chord $= r^{2} - d^{2} = 1.69 - 1.44 = 0.25 = 0.5$ m.
Length inside the rangoli $= 2 \times 0.5 = 1$ m.
A tangent parallel to the line is at distance $1.3$ m from the centre, on either side.
Same side: shift $= 1.3 - 1.2 = 0.1$ m away from the centre.
Other side: shift $= 1.2 + 1.3 = 2.5$ m, across the centre.

Answer: 1 m of the line is inside; shift it 0.1 m outward (or 2.5 m across to the far side) to make it a tangent.

Memory hook: Push the diya line out until the chord shrinks to a dot: that is the tangent.

Number of tangents from a point

Suppose your school draws its circular cycling track on a grid map, with centre at $(0, 0)$ and radius 30 m, each unit being 1 m. Three students stand at $A (18, 24)$ , $B (12, 16)$ and $C (40, 30)$ . Each wants to ride a straight path that just touches the track.

How many such paths can each student take? For the student outside the track, find the length of each path from the student to the point where it touches the track.

Try it first, then show the solutionSolution

$O A^{2} = 1 8^{2} + 2 4^{2} = 324 + 576 = 900$ , so $O A = 30$ m $= r$ : $A$ is on the track, 1 tangent.
$O B^{2} = 144 + 256 = 400$ , so $O B = 20$ m $< 30$ : $B$ is inside, 0 tangents.
$O C^{2} = 1600 + 900 = 2500$ , so $O C = 50$ m $> 30$ : $C$ is outside, 2 tangents.
The radius to the point of contact $T$ is perpendicular to the tangent, so $△ O T C$ is right-angled at $T$ .
$C T^{2} = O C^{2} - r^{2} = 2500 - 900 = 1600$ , so $C T = 40$ m.

Answer: A: 1 path, B: none, C: 2 paths, each 40 m long.

Memory hook: Inside the track no grazing path, on it one, outside two.

Tricks to remember Tangent and Secant of a Circle

Short memory aids. Say each one aloud once, then use it on a question the same day.

0-1-2 by position
In, On, Out gives 0, 1, 2 tangents. The further out you go, the more tangents you get, but never more than 2.
Use it when: Number-of-tangents MCQs and assertion-reason items.
d against r
Less is two, equal is one, more is none: $d < r$ gives 2 common points, $d = r$ gives 1, $d > r$ gives 0.
Use it when: Classifying any line as secant, tangent or non-intersecting.
Secant shrinks to tangent
Picture the chord $2 r^{2} - d^{2}$ getting shorter as $d$ grows. At $d = r$ it becomes a single point.
Use it when: Remembering why a tangent touches at one point and finding chord lengths.

Tangent and Secant of a Circle: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Position of a point and number of tangents · 1 mark · MCQBoard level

A circle has centre $C (2, - 1)$ and radius $5$ units. From which point can exactly one tangent be drawn to the circle?
(a) $(5, 2)$
(b) $(5, 3)$
(c) $(8, - 1)$
(d) $(2, - 1)$

Try it first, then show the solutionSolution

Exactly one tangent means the point lies on the circle, so its distance from $C$ must be $5$ , i.e. distance squared $= 25$ .
(a) $3^{2} + 3^{2} = 18 < 25$ : inside.
(b) $3^{2} + 4^{2} = 25$ : on the circle.
(c) $6^{2} + 0^{2} = 36 > 25$ : outside. (d) is the centre itself: inside.

Answer: (b) $(5, 3)$ , which lies on the circle.

Example 2: Classifying three parallel lines · 3 marks · Short answerHarder board level

A circle has centre $O$ and radius $17$ cm. A line $l$ cuts the circle in a chord of length $30$ cm. Line $m$ is parallel to $l$ , on the opposite side of $O$ , at a distance of $25$ cm from $l$ . Line $n$ is parallel to $l$ , on the same side of $O$ as $l$ , at a distance of $12$ cm from $l$ and further from $O$ . Find the distance of $l$ from $O$ , and state with reason how many points each of $m$ and $n$ has in common with the circle.

Try it first, then show the solutionSolution

The perpendicular from $O$ bisects the chord, so half chord $= 15$ cm. (½)
$d^{2} = 1 7^{2} - 1 5^{2} = 289 - 225 = 64$ , so $l$ is $8$ cm from $O$ . (1)
$m$ is on the other side: its distance from $O$ is $25 - 8 = 17$ cm $= r$ , so $m$ is a tangent with exactly 1 common point. (½ + ½)
$n$ is $8 + 12 = 20$ cm from $O$ , and $20 > 17$ , so $n$ has no common point. (½)

Answer: $l$ is 8 cm from $O$ ; $m$ is a tangent (1 point); $n$ does not meet the circle (0 points).

Example 3: Unknown coordinate, then number of tangents · 2 marks · Very short answerBoard level

A circle has centre $(1, 2)$ and radius $5$ units. The point $P (4, k)$ lies on the circle. Find the possible values of $k$ . Then state, with reason, how many tangents can be drawn to the circle from the point $Q (4, 7)$ .

Try it first, then show the solutionSolution

$P$ on the circle: $(4 - 1)^{2} + (k - 2)^{2} = 25$ , so $(k - 2)^{2} = 16$ . (½)
$k - 2 = \pm 4$ , giving $k = 6$ or $k = - 2$ . (½)
For $Q$ : distance squared $= 3^{2} + 5^{2} = 34$ , and $34 > 25$ . (½)
So $Q$ is outside the circle and exactly 2 tangents can be drawn. (½)

Answer: $k = 6$ or $k = - 2$ ; 2 tangents from $Q (4, 7)$ .

Example 4: Reasoning plus coordinates: secants and tangents · 5 marks · Long answerHarder board level

(i) Explain why no tangent can be drawn to a circle from a point inside it.
(ii) A circle has centre $O (0, 0)$ and radius $10$ units. State the number of tangents from each of $A (6, 8)$ , $B (5, 5)$ and $C (9, 8)$ .
(iii) The line through $A (6, 8)$ and $D (- 6, 8)$ is a secant. Find the length of the chord $A D$ and its distance from $O$ . A tangent to the circle is drawn parallel to $A D$ . Find the two possible distances between this tangent and the chord.

Try it first, then show the solutionSolution

(i) Let $P$ be inside, so $O P < r$ . Any line through $P$ contains $P$ , whose distance from $O$ is less than $r$ . (½)
So the perpendicular distance from $O$ to that line is at most $O P$ , which is less than $r$ . (½)
Hence every line through $P$ cuts the circle at two points, i.e. it is a secant. (½)
So no line through $P$ is a tangent. (½)
(ii) $O A^{2} = 36 + 64 = 100$ : on, 1 tangent. $O B^{2} = 50 < 100$ : inside, 0 tangents. $O C^{2} = 81 + 64 = 145 > 100$ : outside, 2 tangents. (1½)
(iii) $A D$ lies on the line $y = 8$ , so $A D = 6 - (- 6) = 12$ units and its distance from $O$ is $8$ units. Check: $2 100 - 64 = 12$ . (½)
Tangents parallel to $A D$ are at distance $10$ from $O$ : the lines $y = 10$ and $y = - 10$ . (½)
Distances from the chord: $10 - 8 = 2$ units and $8 + 10 = 18$ units. (½)

Answer: (ii) A: 1, B: 0, C: 2. (iii) $A D = 12$ units, 8 units from $O$ ; the tangent is 2 units or 18 units from the chord.

Example 5: Case study: the wall clock · 4 marks · Case-basedHOTS / competency

Suppose a circular wall clock of radius $15$ cm with centre $O$ hangs so that it just rests on a straight horizontal shelf edge below it. A decorative straight thread $24$ cm long is stretched tight across the clock face with both ends on the rim. The thread is parallel to the shelf edge and lies above $O$ . From a nail $N$ on the wall, two strings are pulled tight so that they just touch the rim at $T$ and $T^{'}$ , and each string $N T$ and $N T^{'}$ is $8$ cm long.
(i) What is the shelf edge with respect to the circle, and how far is it from $O$ ? (1 mark)
(ii) How far is the thread from the shelf edge? (1 mark)
(iii) Find $O N$ , and the area of the quadrilateral $O T N T^{'}$ . (2 marks)

Try it first, then show the solutionSolution

(i) The shelf edge meets the rim at exactly one point, so it is a tangent; its distance from $O$ equals the radius, $15$ cm. (1)
(ii) The perpendicular from $O$ bisects the thread, so half thread $= 12$ cm and its distance from $O$ is $1 5^{2} - 1 2^{2} = 81 = 9$ cm. The thread is above $O$ and the shelf is below, so the gap is $9 + 15 = 24$ cm. (½ + ½)
(iii) The radius to the point of contact is perpendicular to the tangent, so $∠ O T N = 9 0^{\circ}$ and $O N^{2} = 1 5^{2} + 8^{2} = 225 + 64 = 289$ , giving $O N = 17$ cm. (1)
$△ O T N$ and $△ O T^{'} N$ are both right-angled with legs $15$ cm and $8$ cm, so area of $O T N T^{'} = 2 \times \frac{1}{2} \times 15 \times 8 = 120$ cm². (1)

Answer: (i) Tangent, 15 cm from $O$ . (ii) 24 cm. (iii) $O N = 17$ cm; area of $O T N T^{'} = 120$ cm².

Example 6: Tangent length, then distances along the line to the centre · 3 marks · Short answerHarder board level

From a point $P$ outside a circle with centre $O$ and radius $7$ cm, a tangent $P T$ of length $24$ cm is drawn, $T$ being the point of contact. Find $O P$ . The line $P O$ meets the circle at $A$ (nearer to $P$ ) and $B$ . Find $P A$ and $P B$ . If $P T^{'}$ is the second tangent from $P$ , find the perimeter of the quadrilateral $O T P T^{'}$ .

Try it first, then show the solutionSolution

The tangent at $T$ is perpendicular to the radius $O T$ , so $∠ O T P = 9 0^{\circ}$ . (½)
$O P^{2} = O T^{2} + P T^{2} = 49 + 576 = 625$ , so $O P = 25$ cm. (1)
$P A = O P - O A = 25 - 7 = 18$ cm and $P B = O P + O B = 25 + 7 = 32$ cm. (1)
Tangents from an external point are equal, so $P T^{'} = 24$ cm; perimeter $= 7 + 24 + 24 + 7 = 62$ cm. (½)

Answer: $O P = 25$ cm, $P A = 18$ cm, $P B = 32$ cm; perimeter of $O T P T^{'} = 62$ cm.

Example 7: Concentric circles: proof and two touching chords · 5 marks · Long answerHarder board level

Two concentric circles have centre $O$ and radii $13$ cm and $5$ cm. A chord $A B$ of the larger circle touches the smaller circle at $P$ .
(a) Prove that $P$ is the midpoint of $A B$ .
(b) Find the length of $A B$ .
(c) Another chord $C D$ of the larger circle is parallel to $A B$ , lies on the other side of $O$ , and also touches the smaller circle, at $Q$ . Find the distance between $A B$ and $C D$ , and the area of the quadrilateral formed by the four points $A$ , $B$ , $D$ , $C$ .

Try it first, then show the solutionSolution

(a) $A B$ is a tangent to the smaller circle at $P$ , so $O P ⊥ A B$ (tangent is perpendicular to the radius at the point of contact). (1)
In $△ O P A$ and $△ O P B$ : $O A = O B$ (radii of the larger circle), $O P$ is common, $∠ O P A = ∠ O P B = 9 0^{\circ}$ . So $△ O P A ≅ △ O P B$ (RHS), and $A P = P B$ . (1)
(b) In right $△ O P A$ : $A P^{2} = O A^{2} - O P^{2} = 1 3^{2} - 5^{2} = 169 - 25 = 144$ , so $A P = 12$ cm and $A B = 2 \times 12 = 24$ cm. (1)
(c) $O Q ⊥ C D$ and $C D ∥ A B$ , so $P$ , $O$ , $Q$ lie on one line perpendicular to both chords; distance between chords $= P Q = 5 + 5 = 10$ cm. (½ + ½)
By the same working as (b), $C D = 24$ cm. (½)
$A B ∥ C D$ , $A B = C D$ and both are perpendicular to $P Q$ , so $A B D C$ is a rectangle of sides $24$ cm and $10$ cm; area $= 240$ cm². (½)

Answer: (b) $A B = 24$ cm. (c) Distance between the chords $= 10$ cm; area $= 240$ cm².

Where students lose marks in Tangent and Secant of a Circle

The slips examiners see most often, and what to write instead.

Mistake: Comparing the distance with the diameter instead of the radius
The question gives the diameter and students forget to halve it.
Wrong working
Diameter 20 cm, $d = 10$ cm, $10 < 20$ , so secant.
Right working
$r = 10$ cm, $d = r$ , so the line is a tangent.
Do this: Write $r = \dots$ as your first line every time.
Usually costs: The full 1 mark in an MCQ, or the classification mark in a 2- or 3-mark question.
Mistake: Saying a point inside has one tangent, or that an outside point has many
Students mix up the number of tangents at the whole circle (infinitely many) with the number from a single point.
Wrong working
From an external point infinitely many tangents can be drawn.
Right working
From an external point exactly two tangents; the circle as a whole has infinitely many (one at each point).
Do this: Learn In-On-Out = 0-1-2 with the reason.
Usually costs: 1 mark in MCQ or assertion-reason items.
Mistake: Forgetting the second parallel tangent
Students draw only the tangent on the near side of the chord.
Wrong working
Chord 8 cm from centre, $r = 10$ : distance to parallel tangent $= 2$ cm only.
Right working
Two tangents: distances $10 - 8 = 2$ cm and $10 + 8 = 18$ cm.
Do this: Whenever 'a tangent parallel to' appears, check both sides of the centre unless the side is stated.
Usually costs: ½ to 1 mark for an incomplete answer.
Mistake: Using the full chord instead of half in Pythagoras
The perpendicular from the centre bisects the chord, but students plug in the whole length.
Wrong working
$r = 13$ , chord 24: $d^{2} = 169 - 576$ , negative.
Right working
Half chord 12: $d^{2} = 169 - 144 = 25$ , $d = 5$ .
Do this: Write 'perpendicular from centre bisects the chord' and halve first.
Usually costs: 1 to 2 marks of working in a 3-mark question.

Exam tips

In MCQs on number of tangents, compare squared distances with $r^{2}$ ; no square roots are needed and you save time.
In assertion-reason items, test A with numbers first, then judge R as a general fact, then check whether R actually explains A.
When you classify a line, always write the comparison ( $d = r$ , $d < r$ or $d > r$ ) as your reason; a bare word like 'tangent' may not get full credit.
Write the reason 'perpendicular from the centre bisects the chord' before any chord-length calculation.
Constructions of tangents are not in the 2026-27 syllabus, so focus on reasoning and calculation, not drawing steps.

Practice questions on Tangent and Secant of a Circle

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A secant $l$ is drawn to a circle. How many tangents to the circle are parallel to $l$ ?
(a) 0
(b) 1
(c) 2
(d) infinitely many
Hint
A tangent in a fixed direction must be at distance $r$ from the centre. On how many sides of the centre?
Answer
(c) 2, one on each side of the centre, each at distance $r$ .
Q2 · 1 mark · Assertion-reasonBoard level
A circle has centre at the origin and radius 6 units.
Assertion (A): Exactly two tangents can be drawn to the circle from the point $(3, 4)$ .
Reason (R): No tangent can be drawn to a circle from a point lying inside it.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Find the distance of $(3, 4)$ from the origin and compare with 6.
Answer
(d). Distance $= 9 + 16 = 5 < 6$ , so the point is inside and no tangent exists: A is false. R is a true fact.
Q3 · 3 marks · Short answerHarder board level
A circle with centre $(3, - 2)$ passes through $(7, 1)$ . Find its radius. Then state, with reason, the number of tangents that can be drawn from $(9, 1)$ and from $(4, 0)$ .
Hint
Radius is the distance from the centre to the given point. Compare squared distances with $r^{2}$ .
Answer
$r^{2} = 16 + 9 = 25$ , $r = 5$ units (1). For $(9, 1)$ : $36 + 9 = 45 > 25$ , outside, 2 tangents (1). For $(4, 0)$ : $1 + 4 = 5 < 25$ , inside, no tangent (1).
Q4 · 3 marks · Short answerHarder board level
In a circle of radius $25$ cm, two parallel chords of lengths $48$ cm and $14$ cm lie on opposite sides of the centre. Find the distance between the chords. A tangent parallel to the chords is drawn on the side of the $14$ cm chord. Find its distance from each chord.
Hint
Use $d = r^{2} - (half chord)^{2}$ for each chord. The tangent is 25 cm from the centre.
Answer
48 cm chord: $d = 625 - 576 = 7$ cm. 14 cm chord: $d = 625 - 49 = 24$ cm (1). Distance between chords $= 7 + 24 = 31$ cm (1). Tangent to 14 cm chord: $25 - 24 = 1$ cm; to 48 cm chord: $25 + 7 = 32$ cm (1).
Q5 · 2 marks · Short answerBoard level
A student writes: "A circle has diameter 14 cm. A line is at a distance of 7 cm from its centre. Since $7 < 14$ , the line is a secant and cuts the circle at two points." Find the error, correct it, and say how many tangents to the circle can be drawn from the foot of the perpendicular from the centre to this line.
Hint
Which quantity should $d$ be compared with: diameter or radius?
Answer
The error is comparing $d$ with the diameter (½). The radius is 7 cm, so $d = r$ and the line is a tangent with exactly one common point (1). The foot of the perpendicular is the point of contact, which lies on the circle, so exactly one tangent can be drawn from it (½).
Q6 · 1 mark · Assertion-reasonBoard level
A circle has centre $O (2, 3)$ and passes through the point $(5, 7)$ .
Assertion (A): Exactly two tangents can be drawn to the circle from the point $P (6, 6)$ .
Reason (R): The tangent at any point of a circle is perpendicular to the radius through the point of contact.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
First find $r^{2}$ from the given point, then find $O P^{2}$ and compare.
Answer
(d). $r^{2} = 3^{2} + 4^{2} = 25$ . $O P^{2} = 4^{2} + 3^{2} = 25$ , so $P$ lies on the circle and only one tangent can be drawn from it: A is false. R is the true theorem.
Q7 · 3 marks · Short answerHarder board level
$P Q$ is a tangent to a circle with centre $O$ at the point of contact $P$ . $O Q = 13$ cm and $P Q = 12$ cm. Find the radius of the circle. A line through $Q$ is at a perpendicular distance of $3$ cm from $O$ ; find the length of the chord it cuts from the circle. Would a line through $Q$ at a distance of $6$ cm from $O$ meet the circle? Give a reason.
Hint
$O P ⊥ P Q$ , so use Pythagoras in $△ O P Q$ . Then use chord $= 2 r^{2} - d^{2}$ .
Answer
$O P ⊥ P Q$ (tangent is perpendicular to radius), so $r^{2} = 1 3^{2} - 1 2^{2} = 25$ , $r = 5$ cm (1). Chord $= 2 25 - 9 = 2 \times 4 = 8$ cm (1). At distance $6$ cm, $6 > 5$ , so the line does not meet the circle (1).
Q8 · 4 marks · Case-basedHOTS / competency
Case study: Imagine a circular garden of radius $5$ m with a circular fountain pool of radius $3$ m at its centre $O$ . A straight path $A B$ runs from one point $A$ on the garden's boundary to another point $B$ on it, and just touches the edge of the pool at $P$ . A second straight path from $A$ also just touches the pool, at $R$ .
(i) Why is $O P$ perpendicular to the path $A B$ ? (1 mark)
(ii) Find the length of the path $A B$ . (1 mark)
(iii) Find the length $A R$ , and the area of the quadrilateral $O P A R$ . (2 marks)
Hint
The path is a tangent to the pool. The perpendicular from the centre bisects a chord of the garden. Tangents from $A$ to the pool are equal.
Answer
(i) $A B$ is a tangent to the pool at $P$ , and the tangent is perpendicular to the radius $O P$ at the point of contact (1). (ii) $A P^{2} = 5^{2} - 3^{2} = 16$ , $A P = 4$ m; $O P ⊥ A B$ bisects the chord, so $A B = 8$ m (1). (iii) Tangents from $A$ are equal, so $A R = A P = 4$ m (1). Area of $O P A R = 2 \times \frac{1}{2} \times 3 \times 4 = 12$ m² (1).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is a secant?Show answer
A line that cuts a circle at two distinct points.
2.What is the point of contact?Show answer
The single common point of a circle and its tangent.
3.How many tangents from a point inside, on and outside a circle?Show answer
0, 1 and 2 respectively.
4.If $d = r$ , what is the line?Show answer
A tangent to the circle.
5.How many tangents does a circle have in all?Show answer
Infinitely many, one at each point.
6.Chord length at distance $d$ from the centre?Show answer
$2 r^{2} - d^{2}$

Tangent and Secant of a Circle: frequently asked questions

What is the difference between a tangent and a secant?

A secant cuts the circle at two points, so part of it lies inside the circle. A tangent only touches the circle at one point and stays outside everywhere else. A tangent can be seen as the limiting case of a secant whose two points merge.

Why can't we draw a tangent from a point inside a circle?

Every line through an inside point passes through a point closer to the centre than the radius. Such a line must cross the circle twice, so it is always a secant and never a tangent.

Can a tangent touch a circle at two points?

No. By definition a tangent has exactly one common point with the circle. A line with two common points is a secant.

Are constructions of tangents in the 2026-27 syllabus?

No. Constructions of tangents were removed and are not in the 2026-27 syllabus.

How do I check where a point lies using coordinates?

Find the distance of the point from the centre using the distance formula and compare it with the radius. Less means inside, equal means on the circle, and greater means outside.