SSC-Standard Mathematics for Forest Posts · Chapter 6

Geometry II: Quadrilaterals, Circles, Tangents and Loci

What to remember

The four angles of a quadrilateral add up to 360°. The interior angles of an n-sided polygon add up to (n − 2) × 180°, and the exterior angles of any convex polygon add up to 360°.
The angle at the centre of a circle is twice the angle at the circumference on the same arc. Opposite angles of a cyclic quadrilateral are supplementary. A tangent is perpendicular to the radius at the point of contact.
Two tangents from an outside point are equal. For intersecting chords PA × PB = PC × PD, and for a tangent and secant PT² = PA × PB.

Quadrilaterals

A quadrilateral has four sides and four angles that add up to 360°.

Figure	Key properties
Parallelogram	Opposite sides parallel and equal; opposite angles equal; adjacent angles supplementary; diagonals bisect each other
Rectangle	Parallelogram with all angles 90°; diagonals equal and bisect each other
Rhombus	Parallelogram with all sides equal; diagonals bisect each other at right angles; diagonals bisect the angles
Square	Rectangle and rhombus together; diagonals equal, perpendicular and bisect each other
Trapezium	One pair of parallel sides; the line joining the midpoints of the non-parallel sides is parallel to them and equals half their sum
Kite	Two pairs of adjacent equal sides; diagonals meet at right angles

Every rhombus is a parallelogram, but not every parallelogram is a rhombus. A parallelogram inscribed in a circle must be a rectangle.

Worked examples.

A parallelogram has one angle of 70°, so its adjacent angle is 110°. If its angles are (2x + 10)° and (3x − 10)° and are adjacent, 5x = 180, x = 36 and the first angle is 82°.
A rhombus with diagonals 16 and 12 has half-diagonals 8 and 6, so its side is √(64 + 36) = 10.
A square of side a has diagonal a√2. A rectangle with sides 6 and 8 has diagonal 10.
In a trapezium with parallel sides 8 and 14, the midline is (8 + 14)/2 = 11.

Polygons

Quantity	Formula
Sum of interior angles	(n − 2) × 180°
Sum of exterior angles	360°
Each interior angle of a regular polygon	(n − 2) × 180° / n
Each exterior angle of a regular polygon	360° / n
Number of diagonals	n(n − 3) / 2

A hexagon has interior angle sum 720°. A regular octagon has each interior angle 135°. A regular polygon with exterior angle 36° has n = 10 sides.
A pentagon has 5 diagonals and an octagon has 8 × 5 / 2 = 20.
Interior angle and exterior angle at one vertex add up to 180°.

Circles: chords and angles

Basic terms: radius, chord, diameter (the longest chord, equal to 2r), arc, segment and sector.

Chord properties.

The perpendicular from the centre to a chord bisects the chord. For chord 24 and radius 13, the distance from the centre is √(169 − 144) = 5.
Equal chords are equally distant from the centre, and subtend equal angles at the centre.
A chord equal to the radius subtends 60° at the centre and 30° at a point on the major arc.

Angle properties.

The angle at the centre is twice the angle at any point on the remaining part of the circle. An angle of 80° at the centre gives 40° at the circumference.
Angles in the same segment are equal.
The angle in a semicircle is 90°.

Cyclic quadrilateral (all four vertices on a circle).

Opposite angles are supplementary. If ∠A = 75°, then ∠C = 105°.
An exterior angle equals the interior opposite angle. An exterior angle of 100° means the interior opposite angle is 100°.

Tangents

A tangent touches the circle at exactly one point. From a point inside the circle there are no tangents, from a point on the circle there is one, and from an outside point there are two.

A tangent is perpendicular to the radius through the point of contact. If the centre is 13 from an outside point P and the radius is 5, the tangent length is √(169 − 25) = 12.
Tangents from one outside point are equal in length. If PA and PB are the tangents and ∠AOB = 110°, then ∠APB = 180° − 110° = 70°, because OAPB is a quadrilateral with two right angles.
Alternate segment theorem. The angle between a tangent and a chord at the point of contact equals the angle in the alternate segment. A 50° tangent-chord angle gives 50° in the alternate segment.
A quadrilateral that surrounds a circle has equal sums of opposite sides: AB + CD = BC + DA. If AB = 7, CD = 9 and BC = 8, then DA = 8.

Lengths with a point outside.

Situation	Rule	Example
Two chords cut at P	PA × PB = PC × PD	4 × 6 = 3 × x gives x = 8
Secant PAB and tangent PT	PT² = PA × PB	PT = 6, PA = 4 gives PB = 9 and chord AB = 5

Common tangents to two circles with radii r₁, r₂ and centre distance d:

Direct (outer) tangent length = √(d² − (r₁ − r₂)²). For d = 13, radii 8 and 3, length = √(169 − 25) = 12.
Transverse (inner) tangent length = √(d² − (r₁ + r₂)²). For d = 17, radii 5 and 3, length = √(289 − 64) = 15.

Position of the two circles	Number of common tangents
Separate (outside each other)	4
Touching externally	3
Intersecting at two points	2
Touching internally	1
One inside the other, no touching	0

Loci

A locus is the path of a point that moves under a given condition.

Condition	Locus
At a fixed distance from a fixed point	A circle with that point as centre
Equidistant from two fixed points	The perpendicular bisector of the segment joining them
Equidistant from two intersecting lines	The pair of angle bisectors of the lines
At a fixed distance from a fixed line	Two lines parallel to the given line, one on each side
Equidistant from two parallel lines	A parallel line midway between them

Area facts linked to quadrilaterals

Parallelograms on the same base and between the same parallels have equal areas. The area of a parallelogram is base × height.
The diagonals of a parallelogram divide it into four triangles of equal area. Each diagonal divides it into two congruent triangles.
The area of a rhombus is ½ × d₁ × d₂. The area of a kite is also ½ × d₁ × d₂.
For a trapezium, area = ½ × (sum of the parallel sides) × height.

Steps to solve circle problems

1. Mark the centre and join it to the points of contact or the chord ends. The radii formed are equal.
2. Look for right angles: a radius meets a tangent at 90°, the diameter makes 90° at the circumference, and a perpendicular from the centre meets a chord at 90°.
3. Use triangles formed by two radii. They are isosceles, so the base angles are equal.
4. For a cyclic quadrilateral, relate opposite angles by the 180° rule and exterior angles by the interior-opposite rule.
5. For a point outside, choose between the tangent-length rule, the tangent-secant rule and the intersecting-chord rule according to what is given.

Example. Two tangents PA and PB are drawn from P, and the radius is 5. If PO = 13, each tangent is 12 and the quadrilateral OAPB has area 2 × (½ × 5 × 12) = 60.

Exam traps

Saying the diagonals of a rectangle are perpendicular. They are equal and bisect each other, but not at right angles unless it is a square.
Saying the diagonals of a rhombus are equal. They are perpendicular, and equal only for a square.
Forgetting that the angle at the centre is twice, not equal to, the angle at the circumference.
Treating the opposite angles of a cyclic quadrilateral as equal. They add up to 180°.
Using PT² = PA × PB with PA as the whole secant. PA is the outer part and PB is the whole secant.
Mixing up the formulas for direct and transverse common tangents (minus versus plus).
Believing a tangent can touch a circle at two points.
Giving the locus of points equidistant from two fixed points as a circle. It is a line.

One-liners

1. The angles of a quadrilateral add up to 360°.
2. The interior angle sum of an n-gon is (n − 2) × 180°.
3. Each exterior angle of a regular n-gon is 360°/n.
4. The diagonals of a rhombus bisect each other at right angles.
5. The diagonals of a rectangle are equal.
6. The angle in a semicircle is 90°.
7. The angle at the centre is twice the angle at the circumference.
8. The opposite angles of a cyclic quadrilateral add up to 180°.
9. A tangent is perpendicular to the radius at the point of contact.
10. Tangents drawn from an outside point are equal.
11. Intersecting chords: PA × PB = PC × PD.
12. The locus of points equidistant from two fixed points is their perpendicular bisector.

Practice questions

The sum of the interior angles of a quadrilateral is
1. 270°
2. 360°
3. 540°
4. 180°
Answer
B. 360°
A quadrilateral splits into two triangles.
One angle of a parallelogram is 70°. An adjacent angle is
1. 20°
2. 70°
3. 290°
4. 110°
Answer
D. 110°
Adjacent angles of a parallelogram are supplementary.
The diagonals of a rhombus are 16 cm and 12 cm. The side of the rhombus is
1. 14 cm
2. 28 cm
3. 10 cm
4. 8 cm
Answer
C. 10 cm
Side = √(8² + 6²) = 10 cm.
The diagonals of a rectangle are
1. equal and bisect each other
2. perpendicular to each other
3. equal and perpendicular
4. unequal but bisect each other
Answer
A. equal and bisect each other
A rectangle has equal diagonals that bisect each other, but not at right angles.
The sum of the interior angles of a hexagon is
1. 1080°
2. 540°
3. 720°
4. 900°
Answer
C. 720°
(6 − 2) × 180° = 720°.
Each interior angle of a regular octagon is
1. 120°
2. 135°
3. 108°
4. 150°
Answer
B. 135°
(8 − 2) × 180°/8 = 135°.
Each exterior angle of a regular polygon is 36°. The number of sides is
1. 9
2. 12
3. 8
4. 10
Answer
D. 10
n = 360/36 = 10.
The number of diagonals of an octagon is
1. 20
2. 24
3. 16
4. 28
Answer
A. 20
n(n − 3)/2 = 8 × 5/2 = 20.
The parallel sides of a trapezium are 8 cm and 14 cm. The line joining the midpoints of the non-parallel sides is
1. 6 cm
2. 22 cm
3. 11 cm
4. 12 cm
Answer
C. 11 cm
The midline equals half the sum of the parallel sides.
Which quadrilateral has diagonals that are always perpendicular but not necessarily equal?
1. Rectangle
2. Parallelogram
3. Trapezium
4. Rhombus
Answer
D. Rhombus
A rhombus has perpendicular diagonals; a rectangle has equal ones.
The angle at the centre of a circle subtended by an arc is 80°. The angle at the circumference on the same arc is
1. 160°
2. 40°
3. 80°
4. 20°
Answer
B. 40°
The angle at the centre is twice the angle at the circumference.
The angle in a semicircle is
1. 60°
2. 90°
3. 45°
4. 180°
Answer
B. 90°
The diameter subtends a right angle at the circumference.
In a cyclic quadrilateral ABCD, ∠A = 75°. Then ∠C is
1. 285°
2. 75°
3. 105°
4. 115°
Answer
C. 105°
Opposite angles of a cyclic quadrilateral are supplementary.
A chord of length 24 cm lies in a circle of radius 13 cm. The distance of the chord from the centre is
1. 12 cm
2. 7 cm
3. 10 cm
4. 5 cm
Answer
D. 5 cm
√(13² − 12²) = 5 cm.
A point P is 13 cm from the centre of a circle of radius 5 cm. The length of the tangent from P is
1. 12 cm
2. √194 cm
3. 18 cm
4. 8 cm
Answer
A. 12 cm
Tangent length = √(169 − 25) = 12 cm.
Tangents PA and PB are drawn from P to a circle with centre O. If ∠AOB = 110°, then ∠APB is
1. 110°
2. 70°
3. 90°
4. 55°
Answer
B. 70°
In quadrilateral OAPB, two angles are 90°, so ∠APB = 180° − 110° = 70°.
From a point P, a tangent PT of length 6 cm and a secant PAB (A nearer to P) are drawn to a circle. If PA = 4 cm, the chord AB is
1. 5 cm
2. 9 cm
3. 4 cm
4. 2 cm
Answer
A. 5 cm
PT² = PA × PB gives PB = 36/4 = 9, so AB = 9 − 4 = 5 cm.
Two chords AB and CD of a circle intersect at P. If PA = 4, PB = 6 and PC = 3, then PD is
1. 8
2. 2
3. 4.5
4. 12
Answer
A. 8
PA × PB = PC × PD gives 24 = 3 × PD, so PD = 8.
The distance between the centres of two circles of radii 8 cm and 3 cm is 13 cm. The length of a direct common tangent is
1. √194 cm
2. 15 cm
3. 10 cm
4. 12 cm
Answer
D. 12 cm
√(13² − (8 − 3)²) = √144 = 12 cm.
The distance between the centres of two circles of radii 5 cm and 3 cm is 17 cm. The length of a transverse common tangent is
1. 12 cm
2. 16 cm
3. 15 cm
4. √285 cm
Answer
C. 15 cm
√(17² − (5 + 3)²) = √225 = 15 cm.
The number of common tangents to two circles that intersect at two points is
1. 2
2. 1
3. 3
4. 4
Answer
A. 2
Intersecting circles have only the two direct common tangents.
The locus of a point equidistant from two fixed points is
1. the angle bisector
2. a line parallel to the line joining them
3. a circle
4. the perpendicular bisector of the line joining them
Answer
D. the perpendicular bisector of the line joining them
Every point on the perpendicular bisector is equidistant from both points.
The locus of a point at a fixed distance from a fixed point is
1. a pair of parallel lines
2. a straight line
3. a circle
4. an ellipse
Answer
C. a circle
The fixed point is the centre and the distance is the radius.
The locus of points equidistant from two intersecting lines is
1. a circle
2. the pair of angle bisectors
3. the perpendicular bisector
4. a single parallel line
Answer
B. the pair of angle bisectors
The two bisectors of the angles formed by the lines.
The locus of points at a fixed distance from a given straight line is
1. one parallel line
2. a perpendicular line
3. two parallel lines
4. a circle
Answer
C. two parallel lines
There is one parallel line on each side.
The angle between a tangent and a chord at the point of contact is 50°. The angle in the alternate segment is
1. 130°
2. 25°
3. 100°
4. 50°
Answer
D. 50°
By the alternate segment theorem the two angles are equal.
In a parallelogram ABCD, ∠A = (2x + 10)° and the adjacent ∠B = (3x − 10)°. Then ∠A is
1. 36°
2. 82°
3. 98°
4. 72°
Answer
B. 82°
5x = 180 gives x = 36, so ∠A = 82°.
A quadrilateral ABCD surrounds a circle. If AB = 7, CD = 9 and BC = 8, then DA is
1. 8
2. 6
3. 7
4. 10
Answer
A. 8
AB + CD = BC + DA gives 16 = 8 + DA.
The exterior angle of a cyclic quadrilateral is 100°. The interior opposite angle is
1. 80°
2. 50°
3. 180°
4. 100°
Answer
D. 100°
The exterior angle equals the interior opposite angle.
A chord equal to the radius of a circle subtends what angle at a point on the major arc?
1. 90°
2. 60°
3. 30°
4. 120°
Answer
C. 30°
The chord subtends 60° at the centre, so 30° on the major arc.
The number of tangents that can be drawn to a circle from a point outside it is
1. 1
2. 2
3. 0
4. 4
Answer
B. 2
Two equal tangents can be drawn from an outside point.
A parallelogram that can be inscribed in a circle must be a
1. kite
2. rectangle
3. rhombus
4. trapezium
Answer
B. rectangle
Opposite angles must be supplementary, and equal, so each is 90°.
Which of the statements is/are correct? 1. Every rhombus is a parallelogram. 2. Every parallelogram is a rhombus.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A parallelogram needs all sides equal to be a rhombus.
Which of the statements is/are correct? 1. The diagonals of a rectangle bisect each other at right angles. 2. The diagonals of a rhombus are equal.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
The rectangle's diagonals are equal; the rhombus's diagonals are perpendicular.
Which of the statements is/are correct? 1. Opposite angles of a cyclic quadrilateral are supplementary. 2. The angle in a semicircle is a right angle.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard circle theorems.
Which of the statements is/are correct? 1. A tangent to a circle can touch it at two points. 2. A tangent is perpendicular to the radius at the point of contact.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
A tangent touches a circle at exactly one point.
Which of the statements is/are correct? 1. Tangents drawn from an external point to a circle are equal in length. 2. A secant touches a circle at one point only.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A secant cuts the circle at two points.
Which of the statements is/are correct? 1. The sum of the exterior angles of any convex polygon is 360°. 2. The sum of the interior angles of an n-sided polygon is (n − 2) × 180°.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard polygon results.
Which of the statements is/are correct? 1. Equal chords of a circle are equidistant from the centre. 2. The perpendicular from the centre to a chord does not bisect the chord.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
The perpendicular from the centre always bisects the chord.
Which of the statements is/are correct? 1. The diagonals of a kite meet at right angles. 2. The diagonals of a parallelogram are always equal.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Equal diagonals hold for a rectangle but not for every parallelogram.
Match the locus condition with the locus: (a) fixed distance from a point, (b) equidistant from two points, (c) equidistant from two intersecting lines. The correct matching is
1. a-circle, b-angle bisector, c-perpendicular bisector
2. a-perpendicular bisector, b-circle, c-angle bisectors
3. a-parallel lines, b-circle, c-perpendicular bisector
4. a-circle, b-perpendicular bisector, c-angle bisectors
Answer
D. a-circle, b-perpendicular bisector, c-angle bisectors
These are the three standard loci.
Match the number of common tangents: (a) circles touching externally, (b) circles touching internally, (c) separate circles. The correct values in order are
1. 1, 3, 4
2. 3, 1, 4
3. 2, 1, 3
4. 4, 3, 1
Answer
B. 3, 1, 4
External contact gives 3, internal contact 1 and separate circles 4.
The longest chord of a circle is
1. any chord through the arc midpoint
2. a tangent
3. the diameter
4. the radius
Answer
C. the diameter
The diameter has length 2r, the maximum possible for a chord.
A rectangle has sides 6 cm and 8 cm. The length of its diagonal is
1. 12 cm
2. 14 cm
3. √28 cm
4. 10 cm
Answer
D. 10 cm
√(36 + 64) = 10 cm.
The diagonal of a square of side 7 cm is
1. 7 cm
2. 7√2 cm
3. 49√2 cm
4. 14 cm
Answer
B. 7√2 cm
The diagonal of a square is a√2.

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1.The area of quadrilateral is 336 m 2 and the perpendiculars drawn to one diagonal from the opposite vertices are 16 m and 12 m long. Find the length of this diagonal
2.The base of a parallelogram is twice as long as its corrosponding height. If the area of the parallelogram is 144 cm², find the mentioned height
3.The given pie chart shows the percentage wise distribution of books in 5 different states A, B, C, D and E. The total number of books is 2500.
4.The Earth makes a complete rotation about its axis in 24 hours. Through what angle will it turn in 3 hours 30 minutes?
5.The ratio of measures of the angles of a triangle is given as 3: 4: 8. What is the measure of the smallest of the three angles?
6.The radii of two circles are 4 cm and 7 cm and the distance between their centres is 12 cm. Find the length of the direct common tangent to the circles.
7.The ratio in which a transverse common tangent drawn to two circles with radii 4 cm and 6 cm, respectively, divides the line joining their centres is:
8.The length of the tangent drawn to a circle of radius 12 cm from a point 18 cm away from the centre of the circle is:
9.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?
10.RA and RB are two tangents from a common point 'R' outside the circle with centre 'O', having a radius of 6 cm. If ARB = 70°, find the measure of the angle AOB.

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