Maharashtra State Board · Class 10 · Mathematics · Geometry Chapter 3

Circle — Formula Sheet

Board Formulas

17 formulas

1.Tangent Theorem
$O P ⊥ l (l tangent at P)$
$O$ : Centre of the circle · $P$ : Point of contact of tangent l
A tangent at any point of a circle is perpendicular to the radius through the point of contact. Converse: a line through the end of a radius and perpendicular to it is a tangent.
2.Tangent Segment Theorem★
$P A = P B$
$P A, P B$ : Tangent segments from external point P touching the circle at A and B
Tangent segments drawn from an external point P to a circle are congruent. Also, OP bisects ∠APB.
3.Length of a Tangent Segment
$P A = O P^{2} - r^{2}$
$O P$ : Distance of the external point from the centre (cm) · $r$ : Radius (cm)
From the right angle at A in ΔOAP (tangent theorem) and Pythagoras. OP is the distance of P from the centre.
4.Touching Circles
$d = r_{1} + r_{2} (externally), d = r_{1} - r_{2} (internally, r_{1} > r_{2})$
$d$ : Distance between the centres (cm) · $r_{1}, r_{2}$ : Radii of the two circles (cm)
If two circles touch each other, their point of contact lies on the line joining their centres. That is why the distances simply add or subtract.
5.Measure of an Arc
$m (minor arc A B) = ∠ A O B, m (major arc A B) = 36 0^{\circ} - ∠ A O B$
$∠ A O B$ : Central angle subtended by the arc
Arc measure equals the central angle. A semicircle measures 180°.
6.Sum of Measures of Arcs
$m (arc A B C) = m (arc A B) + m (arc B C)$
Arcs that share only an endpoint B add up. Useful for finding an unknown arc when the whole circle (360°) is split into pieces.
7.Congruent Arcs and Chords
$arc A B ≅ arc C D \Leftrightarrow chord A B ≅ chord C D$
In the same circle (or congruent circles), chords of congruent arcs are congruent, and conversely.
8.Inscribed Angle Theorem★
$∠ A B C = \frac{1}{2} m (arc A X C)$
The measure of an inscribed angle is half the measure of the arc it intercepts. B is on the circle; arc AXC is the arc inside the angle.
9.Corollaries of the Inscribed Angle Theorem
$∠ A C B = ∠ A D B (same arc), angle in a semicircle = 9 0^{\circ}$
Angles inscribed in the same arc are congruent. An angle inscribed in a semicircle is a right angle, because its arc is 180°.
10.Cyclic Quadrilateral Theorem★
$∠ A + ∠ C = 18 0^{\circ}, ∠ B + ∠ D = 18 0^{\circ}$
Opposite angles of a cyclic quadrilateral are supplementary. Converse: if a pair of opposite angles is supplementary, the quadrilateral is cyclic.
11.Exterior Angle of a Cyclic Quadrilateral
$∠ C B E = ∠ A D C (E on A B produced)$
An exterior angle of a cyclic quadrilateral is congruent to the interior angle opposite to its adjacent interior angle. Both equal 180° − ∠ABC.
12.Tangent-Secant Angle Theorem
$∠ A B C = \frac{1}{2} m (arc A X B)$
Vertex B on the circle, ray BC tangent at B and ray BA a secant (chord BA). The angle is half the arc it intercepts, arc AXB, which lies inside the angle.
13.Angle between Chords Intersecting Inside
$∠ A E C = \frac{1}{2} [m (arc A C) + m (arc B D)]$
Chords AB and CD intersect at E inside the circle. Arc AC is intercepted by ∠AEC and arc BD by its vertically opposite angle ∠BED. Proved by joining a pair of endpoints and using the exterior angle of a triangle.
14.Angle between Secants Intersecting Outside
$∠ A E C = \frac{1}{2} [m (arc A C) - m (arc B D)]$
Secants EBA and EDC meet at E outside the circle, with B and D the nearer points. Half the DIFFERENCE of the far arc and the near arc.
15.Theorem of Internal Division of Chords★
$A E \times E B = C E \times E D$
$E$ : Point of intersection of chords AB and CD (inside the circle)
Chords AB and CD intersect at E inside the circle. The product of the two parts of one chord equals that of the other.
16.Theorem of External Division of Chords
$A E \times B E = C E \times D E$
$E$ : Point of intersection of the secants (outside the circle)
The lines containing chords AB and CD intersect at E outside the circle. Each length is measured from E: AE and BE are both measured along the same secant.
17.Tangent-Secant Segments Theorem★
$E A \times E B = E T^{2}$
$E T$ : Tangent segment from E · $E A, E B$ : Distances from E to the near and far points of the secant
From an external point E, a secant meets the circle at A and B, and a tangent touches it at T. The product of the secant segments equals the square of the tangent segment.