CBSE · Class 10 · Mathematics · Chapter 10

Circles — Formula Sheet

Board Formulas

14 formulas

1.Line and Circle: Three Cases
$common points = 0 (non-intersecting), 1 (tangent), 2 (secant)$
A tangent meets the circle at exactly one point, called the point of contact. A tangent is the limiting case of a secant when its two points of intersection coincide.
2.Number of Tangents from a Point
$P inside: 0, P on the circle: 1, P outside: 2$
No tangent passes through a point inside the circle. There is exactly one tangent at each point of the circle, and exactly two tangents from a point outside it.
3.Tangent Perpendicular to Radius (Theorem 10.1)★
$O A ⊥ X Y, ∠ O A X = ∠ O A Y = 9 0^{\circ}$
$O$ : Centre of the circle · $A$ : Point of contact · $X Y$ : Tangent at A
XY is the tangent at A and O is the centre. The line through A perpendicular to the tangent passes through the centre.
4.Length of a Tangent★
$P A = O P^{2} - r^{2}$
$P A$ : Length of the tangent from P to the point of contact A (cm) · $O P$ : Distance of the external point P from the centre O (cm) · $r$ : Radius, r = OA (cm)
Triangle OAP is right-angled at the point of contact A, so OP (centre to external point) is the hypotenuse. All lengths in the same unit (cm).
5.Equal Tangents from an External Point (Theorem 10.2)★
$P A = P B$
$P$ : External point · $A, B$ : Points of contact of the two tangents
PA and PB are the two tangents from the external point P, touching the circle at A and B.
6.Centre Lies on the Angle Bisector
$∠ A P O = ∠ B P O, ∠ A O P = ∠ B O P$
From the same congruence ΔOAP ≅ ΔOBP. OP bisects the angle between the tangents and also the angle between the radii OA and OB.
7.Angle Between Tangents and Angle at Centre★
$∠ A P B + ∠ A O B = 18 0^{\circ}$
In quadrilateral OAPB the angles at A and B are 90° each, so the other two angles add up to 360° − 180° = 180°.
8.Angle Between Tangents and the Chord of Contact
$∠ A P B = 2 ∠ O A B$
AB is the chord joining the points of contact. Since PA = PB, ∠PAB = 90° − ½∠APB, and ∠OAB = 90° − ∠PAB = ½∠APB.
9.OP Is the Perpendicular Bisector of the Chord of Contact
$O P ⊥ A B, A M = M B$
M is the point where OP meets AB. Both P (PA = PB) and O (OA = OB) are equidistant from A and B, so OP is the perpendicular bisector of AB.
10.Tangents at the Ends of a Diameter
$tangent at A ∥ tangent at B (A B a diameter)$
Both tangents are perpendicular to the diameter AB, so they are parallel. The distance between two parallel tangents equals the diameter 2r.
11.Chord of the Larger of Two Concentric Circles
$A B = 2 R^{2} - r^{2}$
$R$ : Radius of the larger circle (cm) · $r$ : Radius of the smaller circle (cm)
AB is a chord of the larger circle that touches the smaller circle. The radius of the smaller circle to the point of contact is perpendicular to AB, so the chord is bisected at the point of contact.
12.Quadrilateral Circumscribing a Circle
$A B + C D = A D + B C$
All four sides of ABCD touch the circle. Opposite sides have equal sums. A parallelogram circumscribing a circle is therefore a rhombus.
13.Angles at the Centre for a Circumscribed Quadrilateral
$∠ A O B + ∠ C O D = 18 0^{\circ}, ∠ B O C + ∠ A O D = 18 0^{\circ}$
Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre. Join O to the vertices and points of contact; the eight angles at O pair up equally and total 360°.
14.Triangle Circumscribing a Circle
$A B = x + y, B C = y + z, C A = z + x, ar (△ A B C) = r \cdot s$
$x, y, z$ : Tangent lengths from vertices A, B, C (cm) · $r$ : Radius of the inscribed circle (cm) · $s$ : Semi-perimeter of the triangle (cm)
x, y, z are the tangent lengths from A, B, C. The semi-perimeter is s = x + y + z. Splitting ΔABC into ΔOAB, ΔOBC and ΔOCA gives area = ½ r (AB + BC + CA) = rs. Combine with Heron's formula to find an unknown tangent length.