MathsICSE Class 10 · Constructions

Constructing Tangents to a Circle from an External Point (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

To draw tangents from an external point $P$ to a circle with centre $O$: join $OP$, bisect it at $M$, and draw the circle with centre $M$ and radius $MO$. It cuts the given circle at $A$ and $B$; $PA$ and $PB$ are the tangents, because $\angle OAP = \angle OBP = 90^\circ$ (angles in a semicircle).

A and B lie on the circle with diameter OP, so angle OAP and angle OBP are 90°, which makes PA and PB tangents of equal length.

Illustration drawn for QuestionBuddy.

Understand Constructing Tangents from an External Point

Construction (a) in the CISCE syllabus for Constructions is 'tangents to a circle from an external point'. Use ruler and compasses only.

Steps.

1. Draw the circle with centre $O$ and the given radius $r$ ; mark $P$ with the given $O P$ .
2. Join $O P$ and construct its perpendicular bisector to find the mid-point $M$ .
3. With centre $M$ and radius $M O$ (= $M P$ ), draw a circle. It cuts the given circle at $A$ and $B$ .
4. Join $P A$ and $P B$ . These are the required tangents.
5. Measure $P A$ (and $P B$ ) and write the length.

Why it works. $A$ lies on the circle with diameter $O P$ , so $∠ O A P = 9 0^{\circ}$ (angle in a semicircle). A line perpendicular to a radius at its end on the circle is a tangent, so $P A$ is a tangent; likewise $P B$ .

Checking by calculation. $P A = P B = O P^{2} - r^{2}$ . If your measured length differs by more than about 1 mm, redo the arcs.

Tangent at a point on the circle. At a point $A$ on the circle, the tangent is the perpendicular to $O A$ at $A$ : construct a right angle at $A$ on $O A$ produced.

**Tangents inclined at a given angle $θ$ .** Since $∠ A P B + ∠ A O B = 18 0^{\circ}$ , draw two radii $O A$ and $O B$ with $∠ A O B = 18 0^{\circ} - θ$ (use a compass angle), construct the tangents at $A$ and $B$ (perpendicular to the radii), and let them meet at $P$ .

$θ = 9 0^{\circ}$ : $∠ A O B = 9 0^{\circ}$ and $O A P B$ is a square, so $P A = r$ .
$θ = 6 0^{\circ}$ : $∠ A O B = 12 0^{\circ}$ and $O P = 2 r$ .
$θ = 12 0^{\circ}$ : $∠ A O B = 6 0^{\circ}$ (step off the radius once).

Points to remember. From a point inside the circle no tangent can be drawn; from a point on the circle exactly one; from an external point exactly two, and they are equal.

Constructing Tangents from an External Point: formulas to remember

Tangent length (check): $P A = P B = O P^{2} - r^{2}$; Use to verify a measured length.
Why the construction works: $∠ O A P = 9 0^{\circ}$; Angle in a semicircle on $O P$ .
Tangents at a given angle: $∠ A O B = 18 0^{\circ} - ∠ A P B$; Draw the radii first, then the tangents.
Area of the kite OAPB: $Area = r \times P A$; Two right triangles, each $\frac{1}{2} r \times P A$ .

How to solve Constructing Tangents from an External Point questions

Bisect $O P$ , draw the circle on $O P$ as diameter, join $P$ to the two meeting points, then measure and check.

1Draw the circle and the point $P$ accurately, and join $O P$ .
2Bisect $O P$ with two pairs of arcs; mark $M$ .
3Draw the circle with centre $M$ through $O$ and $P$ ; mark $A$ and $B$ where it meets the given circle.
4Join $P A$ and $P B$ ; extend slightly beyond $A$ and $B$ .
5Measure and state $P A$ ; check with $O P^{2} - r^{2}$ .
6If asked, write the justification: ' $∠ O A P = 9 0^{\circ}$ (angle in a semicircle), so $P A ⊥ O A$ and is a tangent'.

Constructing Tangents from an External Point 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 length

Suppose a rope is stretched from the top of a peg $P$ so that it just touches a round water tank of radius 0.5 m. The peg is 1.3 m from the centre of the tank (in a top view).

How long is the rope from the peg to the point where it touches the tank?

Try it first, then show the solutionSolution

The rope is a tangent, so it is perpendicular to the radius at the point of contact.
Length $= 1. 3^{2} - 0. 5^{2} = 1.44$ .

Answer: 1.2 m

Memory hook: Tangent² = distance² − radius².

Two equal lines of sight

Imagine a camera at $P$ , 2.6 m from the centre of a round pillar of radius 1 m. The edges of the pillar in the picture are the two lines of sight that just touch it.

How far is the camera from each point where a line of sight touches the pillar?

Try it first, then show the solutionSolution

Both lines of sight are tangents from $P$ , so they are equal.
Length $= 2. 6^{2} - 1^{2} = 5.76 = 2.4$ m.

Answer: 2.4 m each

Memory hook: Two tangents from one point: twins.

Tricks to remember Constructing Tangents from an External Point

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

Half, circle, join
Halve $O P$ , draw the circle on $O P$ , join $P$ to the two cuts.
Use it when: Remembering the three key steps.
Semicircle gives the square corner
The circle on $O P$ is there only to make $∠ O A P = 9 0^{\circ}$ .
Use it when: Writing the justification.
Angle swap
Angle between the tangents + angle at the centre $= 18 0^{\circ}$ .
Use it when: Tangents inclined at a given angle.

Constructing Tangents from an External Point: solved examples

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

Example 1: How many tangents? · 1 mark · MCQBoard level

The number of tangents that can be drawn to a circle from a point inside it is:
(a) 0
(b) 1
(c) 2
(d) infinitely many

Try it first, then show the solutionSolution

Every line through an interior point cuts the circle twice, so none is a tangent.

Answer: (a)

Example 2: Construct and check · 3 marks · Short answerBoard level

Draw a circle of radius 3.5 cm. Mark a point $P$ 6 cm from its centre $O$ . Construct the two tangents from $P$ , measure one of them and check by calculation.

Try it first, then show the solutionSolution

Draw the circle; mark $P$ with $O P = 6$ cm. Bisect $O P$ at $M$ ( $O M = 3$ cm).
With centre $M$ and radius 3 cm draw a circle cutting the given circle at $A$ and $B$ . Join $P A$ and $P B$ .
$P A \approx 4.9$ cm. Check: $6^{2} - 3. 5^{2} = 23.75 \approx 4.87$ cm.

Answer: $P A = P B \approx 4.9$ cm

Example 3: Tangents and the kite · 4 marks · Long answerHarder board level

Draw a circle with centre $O$ and radius 2.4 cm and a point $P$ with $O P = 5.1$ cm. Construct the tangents $P A$ and $P B$ .
(a) Measure $P A$ and verify it.
(b) Find the area of quadrilateral $O A P B$ .

Try it first, then show the solutionSolution

Bisect $O P$ at $M$ ( $O M = 2.55$ cm); draw the circle on $O P$ as diameter, cutting the given circle at $A$ and $B$ ; join $P A$ , $P B$ .
(a) $P A \approx 4.5$ cm. Check: $5. 1^{2} - 2. 4^{2} = 26.01 - 5.76 = 20.25 = 4.5$ cm.
(b) $O A P B$ is made of two congruent right triangles, each with legs 2.4 cm and 4.5 cm.
Area $= 2 \times \frac{1}{2} \times 2.4 \times 4.5 = 10.8$ cm².

Answer: (a) 4.5 cm (b) 10.8 cm²

Example 4: Tangents at right angles · 4 marks · Long answerHarder board level

Draw a circle of radius 3 cm with centre $O$ . Construct two tangents to it that are perpendicular to each other, meeting at $P$ . Measure $O P$ and $P A$ , and explain the values.

Try it first, then show the solutionSolution

$∠ A P B = 9 0^{\circ}$ , so $∠ A O B = 18 0^{\circ} - 9 0^{\circ} = 9 0^{\circ}$ . Draw radii $O A$ and $O B$ at right angles (construct $9 0^{\circ}$ at $O$ ).
At $A$ construct the perpendicular to $O A$ , and at $B$ the perpendicular to $O B$ . These tangents meet at $P$ .
$O A P B$ has four right angles and $O A = O B$ , so it is a square: $P A = 3$ cm.
$O P$ is its diagonal: $32 \approx 4.24$ cm, so the measurement is about 4.2 cm.

Answer: $P A = 3$ cm, $O P \approx 4.2$ cm

Example 5: Tangent to the inner of two circles · 4 marks · Long answerHOTS / competency

Draw two concentric circles of radii 2 cm and 4 cm with centre $O$ . Take a point $P$ on the outer circle and construct the tangents from $P$ to the inner circle. Measure one tangent and verify. What do you notice about where the tangents meet the outer circle again?

Try it first, then show the solutionSolution

$O P = 4$ cm. Bisect $O P$ at $M$ ; the circle on $O P$ as diameter cuts the inner circle at $A$ and $B$ ; join $P A$ and $P B$ .
$P A = 4^{2} - 2^{2} = 23 \approx 3.46$ cm; measure about 3.5 cm.
Produce $P A$ to meet the outer circle at $Q$ . $O A ⊥ P Q$ , so $A$ is the mid-point of chord $P Q$ and $P Q = 2 P A \approx 6.9$ cm: the inner circle's tangent is bisected at the point of contact.

Answer: $P A = 23 \approx 3.5$ cm; each tangent chord of the outer circle is bisected at its point of contact.

Example 6: Case study: roads to a circular lake · 4 marks · Case-basedHOTS / competency

A circular lake has radius 400 m. A village $P$ is 500 m from the centre $O$ . Two straight roads from $P$ just touch the lake at $A$ and $B$ .
(a) Using the scale 1 cm : 100 m, describe the construction of the roads.
(b) Find the length of each road from $P$ to the lake.
(c) Find the area enclosed by the two roads and the radii $O A$ and $O B$ .

Try it first, then show the solutionSolution

(a) Draw a circle of radius 4 cm and mark $P$ 5 cm from $O$ . Bisect $O P$ at $M$ , draw the circle with centre $M$ through $O$ , and join $P$ to the points $A$ and $B$ where it cuts the lake.
(b) $P A = 5^{2} - 4^{2} = 3$ cm on the drawing, i.e. 300 m.
(c) Area of $O A P B = 2 \times \frac{1}{2} \times 400 \times 300 = 120000$ m².

Answer: (b) 300 m each (c) 120 000 m²

Where students lose marks in Constructing Tangents from an External Point

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

Mistake: Guessing the tangents by eye
A line drawn to 'just touch' the circle is not a construction.
Wrong working
Placing the ruler against the circle and drawing.
Right working
Bisect $O P$ and use the circle on $O P$ as diameter.
Do this: Show the bisecting arcs and the auxiliary circle.
Usually costs: Most of the construction marks.
Mistake: Radius of the auxiliary circle wrong
The second circle must have $O P$ as diameter, so its radius is $\frac{O P}{2}$ , centred at the mid-point.
Wrong working
Circle with centre $P$ and radius $P O$ .
Right working
Circle with centre $M$ (mid-point of $O P$ ) and radius $M O$ .
Do this: Always find $M$ by bisecting $O P$ first.
Usually costs: The tangents and the measurement.
Mistake: No measurement or check
Questions usually ask for the length; a quick calculation catches a slipped compass.
Wrong working
Tangents drawn, length not written.
Right working
' $P A = 4.9$ cm; check $O P^{2} - r^{2} \approx 4.87$ cm.'
Do this: End with a sentence giving the measured value.
Usually costs: The final accuracy mark.

Exam tips

Keep all arcs, including those used to bisect $O P$ .
Label $O$ , $P$ , $M$ , $A$ and $B$ clearly; examiners in 2025 noted wrongly named vertices in a construction question.
Give lengths to the nearest mm and check with $O P^{2} - r^{2}$ when time allows.
If asked to justify, one line is enough: ' $∠ O A P = 9 0^{\circ}$ (angle in a semicircle), so $P A$ is a tangent'.

Practice questions on Constructing Tangents from an External Point

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
Tangents are drawn to a circle of radius 5 cm from a point 13 cm from its centre. Each tangent is:
(a) 8 cm
(b) 12 cm
(c) 13 cm
(d) 18 cm
Hint
$1 3^{2} - 5^{2}$ .
Answer
(b)
Q2 · 3 marks · Short answerBoard level
Draw a circle of radius 2 cm and mark a point $P$ 5.5 cm from its centre. Construct the tangents from $P$ and measure one of them.
Hint
Bisect $O P$ first.
Answer
$P A \approx 5.1$ cm (check: $30.25 - 4 = 26.25 \approx 5.12$ cm).
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): From a point on a circle, two tangents can be drawn to the circle.
Reason (R): The tangent at any point of a circle is perpendicular to the radius through that point.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
How many lines are perpendicular to $O A$ at $A$ ?
Answer
(b)
Q4 · 4 marks · Long answerHarder board level
Draw a circle with centre $O$ and radius 3 cm, and mark a point $A$ on it. Construct the tangent at $A$ , and mark $P$ on it with $P A = 4$ cm. Now construct the second tangent from $P$ . Measure $O P$ and the second tangent.
Hint
The tangent at $A$ is perpendicular to $O A$ .
Answer
Construct $9 0^{\circ}$ at $A$ on $O A$ for the first tangent. $O P = 9 + 16 = 5$ cm. Bisect $O P$ , draw the circle on $O P$ as diameter to get $B$ ; $P B = 4$ cm, equal to $P A$ .
Q5 · 3 marks · Short answerHarder board level
Draw a circle of radius 3.5 cm. Mark a point $P$ 9.1 cm from its centre. Construct the tangents from $P$ , measure one, and verify by calculation.
Hint
$9. 1^{2} = 82.81$ .
Answer
$P A = 82.81 - 12.25 = 70.56 = 8.4$ cm.
Q6 · 3 marks · Short answerHarder board level
The tangents from a point $P$ to a circle of radius 1.5 cm must each be 2 cm long. Find $O P$ , then construct the circle, the point $P$ and the tangents.
Hint
Work backwards with Pythagoras.
Answer
$O P = 1. 5^{2} + 2^{2} = 2.5$ cm. Draw the circle, mark $P$ at 2.5 cm, bisect $O P$ , draw the circle on $O P$ as diameter and join $P$ to the two meeting points; each tangent measures 2 cm.
Q7 · 4 marks · Long answerHOTS / competency
Draw a circle of radius 3 cm. Construct a pair of tangents to it inclined to each other at $12 0^{\circ}$ . Measure the distance from the centre to their meeting point and the length of each tangent.
Hint
The angle at the centre between the radii to the points of contact is $6 0^{\circ}$ .
Answer
Draw radii $O A$ , $O B$ with $∠ A O B = 6 0^{\circ}$ (step the radius once round the circle). Construct perpendiculars to $O A$ at $A$ and to $O B$ at $B$ , meeting at $P$ . $O P = \frac{2 \times 3}{3} = 23 \approx 3.46$ cm and $P A = 3 \approx 1.73$ cm.

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.Which circle do you draw after bisecting $O P$ ?Show answer
The circle with $O P$ as diameter (centre at the mid-point $M$ ).
2.Why is $P A$ a tangent?Show answer
$∠ O A P = 9 0^{\circ}$ (angle in a semicircle), so $P A ⊥ O A$ .
3. $O P = 15$ cm, $r = 9$ cm. Tangent length?Show answer
12 cm
4.Tangents inclined at $9 0^{\circ}$ : angle $A O B$ ?Show answer
$9 0^{\circ}$ ( $O A P B$ is a square).
5.Number of tangents from a point on the circle?Show answer
One.

Constructing Tangents from an External Point: frequently asked questions

How do you construct tangents from an external point?

Join the centre O to the point P and bisect OP at M. Draw the circle with centre M through O and P. Join P to the two points where this circle cuts the given circle; those lines are the tangents.

Why does the construction give tangents?

The meeting points lie on a circle with OP as diameter, so the angle there between the radius and the line to P is 90° (angle in a semicircle). A line perpendicular to the radius at the circle is a tangent.

How do I check my tangent length?

Use Pythagoras: tangent length = √(OP² − r²). Your measurement should agree to within about a millimetre.