CBSE Class 9 Mathematics Plane Geometry II Notes Set 12

Circle: Basic Terms

A circle is the set of all points in a plane at a fixed distance (the radius) from a fixed point (the centre). A chord joins two points of the circle; a diameter is a chord through the centre and is the longest chord. An arc is a part of the circle, and a segment is the region between a chord and an arc.
A secant is a line cutting the circle in two points and a tangent touches it at exactly one point.

Chords

Equal chords of a circle are at equal distances from the centre, and equal chords subtend equal angles at the centre. The converse of both statements is also true.
The perpendicular from the centre to a chord bisects the chord, and the line joining the centre to the mid-point of a chord is perpendicular to the chord. So a chord of length l at distance d from the centre satisfies r² = d² + (l/2)².
A longer chord lies nearer the centre. Through three non-collinear points there is exactly one circle. When two circles meet in two points, the line of centres is the perpendicular bisector of the common chord.

Angles in a Circle

The angle subtended by an arc at the centre is double the angle it subtends at any point on the remaining part of the circle.
Angles in the same segment of a circle are equal. The angle in a semicircle is a right angle, and conversely the hypotenuse of a right triangle is a diameter of its circumcircle.
Angles subtended at the circumference by equal chords are equal.

Cyclic Quadrilaterals

If all four vertices of a quadrilateral lie on a circle, it is cyclic. Its opposite angles add to 180°, and each exterior angle equals the interior opposite angle.
Converse: if the opposite angles of a quadrilateral are supplementary, the quadrilateral is cyclic. A parallelogram inscribed in a circle must be a rectangle, and a trapezium inscribed in a circle must be isosceles.
Ptolemy's theorem: in a cyclic quadrilateral ABCD, AC × BD = AB × CD + BC × DA.

Illustration 1. A chord of length 16 cm is drawn in a circle of radius 10 cm. Find its distance from the centre.
Answer:
Half the chord = 8 cm, so d = √(10² − 8²) = 6 cm.

Illustration 2. In a cyclic quadrilateral ABCD, ∠A = 3x + 10° and ∠C = 5x − 30°. Find ∠B if ∠D = 100°.
Answer:
Opposite angles add to 180°: 3x + 10 + 5x − 30 = 180, so 8x = 200 and x = 25. This gives ∠A = 85°, ∠C = 95°.
∠B = 180° − ∠D = 180° − 100° = 80°.

Illustration 3. The angle subtended at the centre of a circle by an arc is 130°. Find the angle it subtends at a point on the major arc.
Answer:
The angle at the circumference is half the angle at the centre: 130° ÷ 2 = 65°.

Tangents

A tangent at a point is perpendicular to the radius through that point. From an external point exactly two tangents can be drawn, and they are equal in length; they make equal angles with the line joining the point to the centre.
If the angle between the two tangents from P is θ, the angle subtended by the two points of contact at the centre is 180° − θ.
Tangent–chord (alternate segment) theorem: the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
A circle inscribed in a quadrilateral forces the sums of opposite sides to be equal (AB + CD = BC + DA). The inradius of a right triangle with legs a, b and hypotenuse c is (a + b − c)/2; in general r = area ÷ semi-perimeter.

Secants and Chord Products

Intersecting chords: if chords AB and CD meet at P inside the circle, PA × PB = PC × PD.
From an external point P, if a secant cuts the circle at A and B and a tangent touches it at T, then PT² = PA × PB. Two secants from P satisfy PA × PB = PC × PD.
The shortest chord through a point inside the circle is the one perpendicular to the line from that point to the centre.

Two Circles

Let the radii be R and r, and the distance between centres d. If the circles touch externally, d = R + r; if internally, d = R − r.
Length of a direct (external) common tangent = √(d² − (R − r)²). Length of a transverse (internal) common tangent = √(d² − (R + r)²).
For circles touching externally, the direct common tangent is 2√(Rr) and the common tangent at the point of contact is perpendicular to the line of centres.

Illustration 4. From a point 17 cm from the centre of a circle of radius 8 cm, a tangent is drawn. Find its length.
Answer:
The radius, tangent and line to the centre form a right triangle: length = √(17² − 8²) = √225 = 15 cm.

Illustration 5. Two chords AB and CD of a circle meet at P inside it. If AP = 6, PB = 4 and CP = 3, find PD.
Answer:
PA × PB = PC × PD gives 24 = 3 × PD, so PD = 8.

Illustration 6. The direct common tangent to two circles of radii 9 cm and 4 cm has length 12 cm. Find the distance between their centres.
Answer:
d = √(12² + (9 − 4)²) = √169 = 13 cm.

Illustration 7. A circle is inscribed in a right triangle with legs 5 cm and 12 cm. Find the inradius.
Answer:
Hypotenuse = 13 cm, so r = (5 + 12 − 13)/2 = 2 cm.

Illustration 8. A tangent PT = 12 cm and a secant PAB from P cuts the circle so that PA = 8 cm. Find AB.
Answer:
PT² = PA × PB, so 144 = 8 × PB and PB = 18. AB = PB − PA = 10 cm.