Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 7

Circles, Chords, Cyclic Quadrilaterals, Tangents and Constructions

What to remember

The perpendicular from the centre bisects a chord; the angle at the centre is twice the angle on the remaining circle; the angle in a semicircle is 90°.
Opposite angles of a cyclic quadrilateral add up to 180°; a tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal.
Constructions use only a ruler and compasses: bisectors, standard angles, tangents, division of a segment in a ratio, and triangles of given data.

Basic terms

A circle is the set of points at a fixed distance (radius) from a fixed point (centre). A chord joins two points of the circle; the diameter is the longest chord, equal to twice the radius. A secant cuts the circle at two points; a tangent touches it at exactly one point. An arc is a part of the circle: the shorter one is the minor arc and the longer is the major arc. A segment is the region between a chord and an arc. A sector is the region between two radii and an arc.

Equal arcs subtend equal angles at the centre, and equal chords cut equal arcs. Congruent circles have equal radii.

Chords

1. The perpendicular from the centre to a chord bisects the chord.
2. The line from the centre to the mid-point of a chord is perpendicular to the chord.
3. Equal chords are equidistant from the centre; chords equidistant from the centre are equal.
4. A longer chord is nearer to the centre.
5. The perpendicular bisector of a chord passes through the centre.
6. Exactly one circle passes through three non-collinear points.

Chord length formula: if a chord is at distance d from the centre of a circle of radius r, its length is 2√(r² − d²). Example: r = 13, d = 5 gives 2 × 12 = 24.

Angles in a circle

Result	Statement
Central angle theorem	angle at the centre = 2 × angle at the remaining circle
Same segment	angles in the same segment are equal
Semicircle	angle in a semicircle = 90°
Converse	if a segment subtends equal angles at two points on the same side, the four points are concyclic

Cyclic quadrilaterals

A quadrilateral whose four vertices lie on a circle is cyclic. Its opposite angles are supplementary (sum 180°), and each exterior angle equals the interior opposite angle. The converse is true: if the opposite angles of a quadrilateral are supplementary, it is cyclic. A parallelogram that is cyclic must be a rectangle; a cyclic rhombus is a square; a cyclic trapezium is isosceles.

Example: ∠A = 100° gives ∠C = 80°. If ∠B = 65°, the exterior angle at D is 65°.

Tangents

Fact	Detail
Perpendicularity	tangent ⟂ radius at the point of contact
Equal tangents	two tangents from an external point have equal lengths
Count	external point: 2; point on circle: 1; interior point: 0
Length	tangent length = √(d² − r²), with d = distance from the centre
Angle sum	angle between tangents + angle at the centre between the contact radii = 180°
Parallel tangents	a circle has two tangents parallel to any given line

Alternate segment theorem: the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.

Intersecting chords: if chords AB and CD meet inside at P, then PA × PB = PC × PD. For a tangent PT and secant PAB from an external point, PT² = PA × PB.

Example: PA = 4, AB = 5 gives PB = 9 and PT = 6.

A quadrilateral circumscribed about a circle has AB + CD = BC + DA. The incircle of a right triangle has radius r = (a + b − c)/2, where c is the hypotenuse. The circumcircle of a right triangle has its centre at the mid-point of the hypotenuse, so R = c/2. For any triangle, inradius = area/semi-perimeter.

Two circles can have 4 common tangents (apart), 3 (touching externally), 2 (intersecting), 1 (touching internally) or 0 (one inside the other). Length of a direct common tangent = √(d² − (r₁ − r₂)²). Length of a transverse common tangent = √(d² − (r₁ + r₂)²).

Worked examples

1. Two tangents from P touch a circle with centre O at A and B, and ∠APB = 60°. In quadrilateral OAPB two angles are 90°, so ∠AOB = 360° − 90° − 90° − 60° = 120°.
2. Chords AB and CD meet at P inside a circle with PA = 4, PB = 6 and PC = 3. Then 4 × 6 = 3 × PD, so PD = 8.
3. A quadrilateral ABCD is drawn around a circle with AB = 7, CD = 5 and BC = 6. Then 7 + 5 = 6 + DA, so DA = 6.
4. The inradius of a right triangle with sides 6, 8 and 10 is (6 + 8 − 10)/2 = 2. Check with area/semi-perimeter = 24/12 = 2.
5. Two circles of radii 8 and 3 have centres 13 apart. The direct common tangent is √(169 − 25) = 12.
6. A chord of length 24 lies in a circle of radius 13. The distance from the centre is √(169 − 144) = 5.

Arc, angle and tangent relations

Arc length and sector area follow from the central angle θ: arc = (θ/360) × 2πr and sector area = (θ/360) × πr². A minor arc subtends a central angle less than 180°. Angles subtended by the same arc at points on the major arc side are equal, and the angle at the minor arc side is supplementary to it. If two circles touch, the point of contact and the two centres are collinear. If two circles intersect, the line joining the centres is the perpendicular bisector of the common chord.

Constructions

1. Perpendicular bisector of a segment: arcs of equal radius (more than half the length) from both ends; join the two crossing points.
2. Angle bisector: arc cutting both arms, then equal arcs from those points.
3. Standard angles: 60° with one arc, 90° by perpendicular, 120° by two arcs, 30° and 45° by bisecting; 75° by bisecting the angle between 60° and 90°; 105° and 135° similarly.
4. Tangent at a point on a circle: draw the radius to the point and erect a perpendicular there.
5. Tangents from an external point P: join P to the centre O, find the mid-point M of OP, draw a circle with centre M and radius MO. It meets the given circle at the contact points T₁ and T₂; PT₁ and PT₂ are the tangents, because the angle in a semicircle is 90°.
6. Dividing a segment in the ratio m : n: draw a ray from A at an acute angle, mark m + n equal parts, join the last mark to B and draw a line parallel to it from the m-th mark. This works by the BPT.
7. Triangle similar to a given triangle with scale factor m/n: the same idea, using parallel lines.
8. Triangles from given data: SSS, SAS, ASA and RHS data all allow construction. Circumcircle: perpendicular bisectors. Incircle: angle bisectors.

For every construction a teacher should insist on a rough sketch, clear steps and a justification.

Classroom angle

Let learners roll a coin along a circle to see the tangent, draw many cyclic quadrilaterals and measure, and fold paper circles to find the centre and diameter. Misconceptions: thinking a tangent can meet the circle twice, mixing up circumcentre and incentre, and using central-angle rules for angles that do not stand on the same arc.

Exam traps

Opposite angles of a cyclic quadrilateral are supplementary, not equal.
A tangent touches at one point; a secant cuts at two.
Angle in a semicircle is 90°, not 180°.
The central angle is twice the inscribed angle, not equal to it.
Circumcentre (perpendicular bisectors) and incentre (angle bisectors) are different.
Centroid is the meet of medians; orthocentre is the meet of altitudes.
Direct common tangent uses the difference of radii; transverse common tangent uses the sum.
Three collinear points lie on no circle.

One-liners

1. The diameter is the longest chord.
2. The tangent is perpendicular to the radius at the point of contact.
3. From an external point exactly two equal tangents can be drawn.
4. Central angle = 2 × inscribed angle on the same arc.
5. Angle in a semicircle = 90°.
6. Cyclic quadrilateral: opposite angles sum to 180°.
7. Tangent length = √(d² − r²).
8. PT² = PA × PB for a tangent and a secant.
9. Intersecting chords: PA × PB = PC × PD.
10. Right triangle circumradius = half the hypotenuse.
11. Perpendicular bisectors meet at the circumcentre.
12. Externally touching circles have 3 common tangents.

Practice questions

A line that touches a circle at exactly one point is called a
1. secant
2. chord
3. diameter
4. tangent
Answer
D. tangent
A tangent meets the circle in one point; a secant meets it in two.
The longest chord of a circle is its
1. diameter
2. tangent
3. radius
4. arc
Answer
A. diameter
The diameter passes through the centre and has length 2r.
The tangent at any point of a circle is perpendicular to
1. every chord of the circle
2. the radius through the point of contact
3. the diameter parallel to it
4. the secant through the centre of another circle
Answer
B. the radius through the point of contact
Tangent ⟂ radius at the point of contact.
The number of tangents that can be drawn to a circle from a point outside it is
1. 1
2. 2
3. 3
4. infinitely many
Answer
B. 2
Two tangents, of equal length, can be drawn from an external point.
The number of tangents to a circle from a point lying on the circle is
1. 0
2. 1
3. 2
4. 4
Answer
B. 1
Only the tangent at that point exists.
The number of tangents to a circle from a point inside it is
1. 1
2. 2
3. 3
4. 0
Answer
D. 0
Every line through an interior point cuts the circle twice.
An angle in a semicircle is
1. a straight angle
2. an obtuse angle
3. a right angle
4. an acute angle
Answer
C. a right angle
The angle subtended by a diameter on the circle is 90°.
The perpendicular from the centre of a circle to a chord
1. is equal to the chord
2. is always a radius equal to half the chord
3. is parallel to the chord
4. bisects the chord
Answer
D. bisects the chord
This is a standard chord theorem.
The angle subtended by an arc at the centre is
1. supplementary to the angle at the circle
2. half the angle at the circle
3. twice the angle it subtends at any point on the remaining part of the circle
4. equal to the angle at the circle
Answer
C. twice the angle it subtends at any point on the remaining part of the circle
Central angle = 2 × inscribed angle.
In a cyclic quadrilateral the opposite angles are
1. supplementary
2. equal
3. complementary
4. both right angles
Answer
A. supplementary
They add up to 180°.
The maximum number of common tangents to two circles is
1. 2
2. 3
3. 6
4. 4
Answer
D. 4
Two separate circles have two direct and two transverse common tangents.
Two circles touching externally have how many common tangents?
1. 1
2. 3
3. 2
4. 4
Answer
B. 3
Two direct tangents and one at the point of contact.
Two circles touching internally have how many common tangents?
1. 2
2. 1
3. 3
4. 0
Answer
B. 1
Only the tangent at the point of contact.
Equal chords of a circle are
1. always diameters
2. at different distances from the centre
3. equidistant from the centre
4. parallel to each other
Answer
C. equidistant from the centre
Equal chords lie at equal distances from the centre.
The angle between a tangent and a chord through the point of contact equals
1. the angle in the alternate segment
2. twice the angle in the same segment
3. the angle at the centre on the same arc
4. 90° always
Answer
A. the angle in the alternate segment
This is the alternate segment theorem.
The radius of a circle is 13 cm and a chord is 5 cm from the centre. The chord's length is
1. 12 cm
2. 18 cm
3. 26 cm
4. 24 cm
Answer
D. 24 cm
Half chord = √(13² − 5²) = 12, so chord = 24.
The length of the tangent from a point 13 cm from the centre of a circle of radius 5 cm is
1. 12 cm
2. 10 cm
3. 18 cm
4. 8 cm
Answer
A. 12 cm
√(13² − 5²) = √144 = 12.
An arc subtends an angle of 80° at the centre. The angle it subtends at a point on the remaining circle is
1. 100°
2. 80°
3. 40°
4. 160°
Answer
C. 40°
The inscribed angle is half the central angle.
In a cyclic quadrilateral ABCD, ∠A = 100°. Then ∠C is
1. 260°
2. 90°
3. 80°
4. 100°
Answer
C. 80°
Opposite angles add to 180°, so ∠C = 80°.
In a cyclic quadrilateral ABCD, ∠B = 65°. The exterior angle at D, formed by extending side CD, equals
1. 65°
2. 25°
3. 115°
4. 130°
Answer
A. 65°
The exterior angle equals the interior opposite angle ∠B.
Two tangents from an external point P touch a circle with centre O at A and B. If ∠APB = 60°, then ∠AOB is
1. 90°
2. 60°
3. 30°
4. 120°
Answer
D. 120°
∠APB + ∠AOB = 180° in quadrilateral OAPB.
Two chords AB and CD of a circle meet at an interior point P. If PA = 4, PB = 6 and PC = 3, then PD is
1. 2
2. 4.5
3. 12
4. 8
Answer
D. 8
PA × PB = PC × PD → 24 = 3 × PD → PD = 8.
From an external point P a tangent PT and a secant PAB are drawn, with PA = 4 cm and AB = 5 cm. Then PT is
1. 6 cm
2. 4.5 cm
3. 20 cm
4. 9 cm
Answer
A. 6 cm
PT² = PA × PB = 4 × 9 = 36.
The radius of the circle passing through the vertices of a right triangle with hypotenuse 10 cm is
1. 2.5 cm
2. 5 cm
3. 10 cm
4. 20 cm
Answer
B. 5 cm
The hypotenuse is a diameter of the circumcircle.
The inradius of a right triangle with sides 6, 8 and 10 cm is
1. 2.4 cm
2. 3 cm
3. 2 cm
4. 4 cm
Answer
C. 2 cm
r = (a + b − c)/2 = (6 + 8 − 10)/2 = 2; or area/s = 24/12 = 2.
Two circles with radii 8 cm and 3 cm have centres 13 cm apart. The length of a direct common tangent is
1. 10 cm
2. 12 cm
3. 11 cm
4. √178 cm
Answer
B. 12 cm
√(13² − (8 − 3)²) = √144 = 12.
Two circles with radii 3 cm and 2 cm have centres 13 cm apart. The length of a transverse common tangent is
1. 13 cm
2. 10 cm
3. 12 cm
4. √165 cm
Answer
C. 12 cm
√(13² − (3 + 2)²) = √144 = 12.
A quadrilateral ABCD is drawn around a circle. If AB = 7, CD = 5 and BC = 6, then DA is
1. 7
2. 5
3. 8
4. 6
Answer
D. 6
AB + CD = BC + DA → 12 = 6 + DA.
The number of tangents to a circle that are parallel to a given line is
1. 1
2. 2
3. 4
4. 0
Answer
B. 2
There are two parallel tangents, at opposite ends of the diameter perpendicular to the line.
To construct a tangent from an external point P to a circle with centre O, a circle is drawn on
1. OP as diameter
2. the tangent as its diameter
3. the radius of the given circle as diameter
4. OP as radius
Answer
A. OP as diameter
The angle in a semicircle is 90°, so the meeting points give OT ⟂ TP.
In the construction of dividing a line segment AB in the ratio 3 : 2, a ray is drawn from A making an acute angle and the number of equal parts marked on it is
1. 5
2. 3
3. 2
4. 6
Answer
A. 5
3 + 2 = 5 equal parts are marked.
A 75° angle can be constructed by bisecting the angle between
1. 90° and 120°
2. 45° and 120°
3. 60° and 90°
4. 30° and 60°
Answer
C. 60° and 90°
(60° + 90°)/2 = 75°.
The perpendicular bisectors of the sides of a triangle meet at the
1. centroid
2. incentre
3. orthocentre
4. circumcentre
Answer
D. circumcentre
This point is equidistant from the three vertices.
The angle bisectors of a triangle meet at the
1. circumcentre
2. excentre
3. orthocentre
4. incentre
Answer
D. incentre
The incentre is the centre of the incircle.
Consider the statements: 1. Tangents drawn from an external point to a circle are equal in length. 2. A tangent to a circle meets it in two points. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A tangent meets the circle at one point only.
Consider the statements: 1. Angles in the same segment of a circle are equal. 2. Opposite angles of a cyclic quadrilateral are equal. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Opposite angles of a cyclic quadrilateral are supplementary.
Consider the statements: 1. The perpendicular from the centre to a chord bisects the chord. 2. The line from the centre to the mid-point of a chord is perpendicular to it. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
A theorem and its converse both hold (for a chord that is not a diameter).
Consider the statements: 1. A parallelogram inscribed in a circle must be a rectangle. 2. A rhombus inscribed in a circle must be a square. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Opposite angles are equal and supplementary, so each is 90°.
Consider the statements: 1. The centre of a circle lies on the perpendicular bisector of every chord. 2. A circle can pass through any three points in a plane. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Three collinear points do not lie on any circle.
Consider the statements: 1. The tangent at a point is the limiting position of a secant as its two intersection points merge. 2. A circle has a tangent parallel to every direction. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are correct: every direction has two parallel tangents.
Which pair is correctly matched?
1. Cyclic quadrilateral : adjacent angles supplementary
2. Alternate segment theorem : tangent–chord angle equals angle in the other segment
3. Tangent length from an external point : two different values
4. Angle in a semicircle : 60°
Answer
B. Alternate segment theorem : tangent–chord angle equals angle in the other segment
Only the first pair is correct.
A teacher wants Class X learners to see why the tangent is perpendicular to the radius. The best approach is to
1. give the proof only in words
2. roll a coin or a ruler along a circle and show the shortest distance from the centre is along the radius
3. ask them to memorise the statement
4. show a diagram of one tangent only
Answer
B. roll a coin or a ruler along a circle and show the shortest distance from the centre is along the radius
A concrete activity leads to the idea; then proof follows.
To explain that a cyclic quadrilateral has supplementary opposite angles, the most effective first step is to
1. avoid using diagrams
2. tell them the rule before any drawing
3. skip the proof for exams
4. have learners draw several cyclic quadrilaterals and measure their angles
Answer
D. have learners draw several cyclic quadrilaterals and measure their angles
Exploration followed by proof develops understanding.
Points A, B, C, D lie on a circle with C and D on the same side of AB. If ∠ACB = 35°, then ∠ADB is
1. 145°
2. 55°
3. 70°
4. 35°
Answer
D. 35°
Angles in the same segment are equal.

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1.In a circle, ABCD is a cyclic quadrilateral in which AE is drawn parallel to CD, and BA is produced to F. If ∠ABC = 85° and ∠FAE = 24°, find the value of ∠BCD.
2.PQ and RS are two parallel chords of a circle of length 10 cm and 24 cm, respectively, and lie on the same side of the centre O. If the distance between the chords is 7 cm, what is the radius (in cm) of the circle?
3.Two parallel chords are drawn in a circle of diameter 10 m. The length of one chord is 8 m and the distance between the two chords is 6 m. Find the length of the other chord.
4.The radii of two concentric circles are 35 cm and 21 cm. If the chord of the greater circle is a tangent to the smaller circle, then the length of that chord is:
5.Consider two concentric circles having radii 17 cm and 15 cm. What is the length (in cm) of the chord, of the bigger circle, which is a tangent to the smaller circle?
6.PQ is a diameter of a circle and PS is a chord. If PQ is 82 cm and PS is 80 cm, the distance (in cm) of PS from the center of the circle is:
7.A chord of 10 cm passes through the centre of a circle. What is the diameter of the circle?
8.Two concentric circles have radii of 15 cm and 17 cm. Find out the length of the chord of the larger circle that is tangent to the smaller circle.
9.Side PQ of a cyclic quadrilateral PQRS is produced to the point T. If , then the measure of is:14
10.In a circle with centre at O and radius 8 cm, AB is a chord of length 14 cm. If OM is perpendicular to AB, then the length of OM is:

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