Page 1 Circles and Areas related to Circles Page 2 Circles and Areas related to Circles Area of seg AxB (Minor) = Area of sector O-AxB – Area of ? OAB O A B B A O x Q Minor segment Major segment x O A B T = 180 Semicircle (O-AxB) & (O-AYB) Y B A O x T Q A O x B Minor sector Major sector Y T O P Radius Distance from the center of the circle to the boundary of the circle. It is denoted by ‘r’ . OP is a radius P O Q Chord Distance between any two points on the circle. A circle has innitely many chords. PQ is a chord A B O Secant A line which intersects the circle in two distinct points. AB is a secant Segment Area enclosed by a chord and the arc. Ex.: segment AxB O « « « « « « « « « « « O L Circumference The length around the boundary of the circle C = 2pr = pd Tangent It is a line that touches the circle at only one point. Sector Area enclosed between two radii and corresponding arc of a circle. B A O Arc A part of the boundary or portion of the circumference. Length of an arc × 2pr = ? 360 × pr 2 ? 360 Area of Major sector Area of the sector = Area of the Major segment = pr 2 - Area of the minor segment Diameter A line segment passing through the center and connecting two points on the boundary of the circle. It is denoted by ‘d’ . AB is a diameter. Diameter is the longest chord in the circle. It is double of radius i.e. d = 2r = Area of circle - Area of the Minor sector Circles & Areas Related to Circles Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center). CIRCLES & AREAS RELATED TO CIRCLES 25 Area of a circle = pr 2 Page 3 Circles and Areas related to Circles Area of seg AxB (Minor) = Area of sector O-AxB – Area of ? OAB O A B B A O x Q Minor segment Major segment x O A B T = 180 Semicircle (O-AxB) & (O-AYB) Y B A O x T Q A O x B Minor sector Major sector Y T O P Radius Distance from the center of the circle to the boundary of the circle. It is denoted by ‘r’ . OP is a radius P O Q Chord Distance between any two points on the circle. A circle has innitely many chords. PQ is a chord A B O Secant A line which intersects the circle in two distinct points. AB is a secant Segment Area enclosed by a chord and the arc. Ex.: segment AxB O « « « « « « « « « « « O L Circumference The length around the boundary of the circle C = 2pr = pd Tangent It is a line that touches the circle at only one point. Sector Area enclosed between two radii and corresponding arc of a circle. B A O Arc A part of the boundary or portion of the circumference. Length of an arc × 2pr = ? 360 × pr 2 ? 360 Area of Major sector Area of the sector = Area of the Major segment = pr 2 - Area of the minor segment Diameter A line segment passing through the center and connecting two points on the boundary of the circle. It is denoted by ‘d’ . AB is a diameter. Diameter is the longest chord in the circle. It is double of radius i.e. d = 2r = Area of circle - Area of the Minor sector Circles & Areas Related to Circles Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center). CIRCLES & AREAS RELATED TO CIRCLES 25 Area of a circle = pr 2 cot ? tan ? sin 2 ? + cos 2 ? = 1 cosec 2 ? - cot 2 ? = 1 sec 2 ? - tan 2 ? = 1 Proof Proof Theorem 2 If two tangents are drawn from an external point outside the circle; the tangents are equal. Construction : Draw a line from point P to the center of the circle and join OT & OQ also. To prove : l (PT) = l (PQ) In ? PQO and ? PTO OQ = OT OP = OP PQO = PTO = 90 o Thus , ? PQO ? PTO PT = PQ l (PT) = l (PQ) ? Common side in both triangles ? Right-angle Hypotenuse Side Rule ? (C.S.C.T) Corresponding Side of Congruent Triangle ? Radius of same circle ? Each 90 o Theorem 1 The tangent at any point of a circle is perpendicular to the radius through the point of contact. To prove : OP is perpendicular to xy i.e OP - xy , OQ > OP If we take any point Q on xy other than P , it will be outside the circle, because if Q lies inside the circle, xy will become a secant and not a tangent, as the tangent touches the circle at a single point. Likewise , if we take any Point Q, other than P , it will lie outside the circle. This proves that OP is the shortest distance between point O and Tangent xy . Therefore, OP - to tangent xy T O Q P = = ? ? T O Q P = = ? ? O P x ? ? O P Q Y x ? ? CIRCLES & AREAS RELATED TO CIRCLES 26 Page 4 Circles and Areas related to Circles Area of seg AxB (Minor) = Area of sector O-AxB – Area of ? OAB O A B B A O x Q Minor segment Major segment x O A B T = 180 Semicircle (O-AxB) & (O-AYB) Y B A O x T Q A O x B Minor sector Major sector Y T O P Radius Distance from the center of the circle to the boundary of the circle. It is denoted by ‘r’ . OP is a radius P O Q Chord Distance between any two points on the circle. A circle has innitely many chords. PQ is a chord A B O Secant A line which intersects the circle in two distinct points. AB is a secant Segment Area enclosed by a chord and the arc. Ex.: segment AxB O « « « « « « « « « « « O L Circumference The length around the boundary of the circle C = 2pr = pd Tangent It is a line that touches the circle at only one point. Sector Area enclosed between two radii and corresponding arc of a circle. B A O Arc A part of the boundary or portion of the circumference. Length of an arc × 2pr = ? 360 × pr 2 ? 360 Area of Major sector Area of the sector = Area of the Major segment = pr 2 - Area of the minor segment Diameter A line segment passing through the center and connecting two points on the boundary of the circle. It is denoted by ‘d’ . AB is a diameter. Diameter is the longest chord in the circle. It is double of radius i.e. d = 2r = Area of circle - Area of the Minor sector Circles & Areas Related to Circles Circle is a collection of all points on a plane which are at constant distance (radius) from a ?xed point (center). CIRCLES & AREAS RELATED TO CIRCLES 25 Area of a circle = pr 2 cot ? tan ? sin 2 ? + cos 2 ? = 1 cosec 2 ? - cot 2 ? = 1 sec 2 ? - tan 2 ? = 1 Proof Proof Theorem 2 If two tangents are drawn from an external point outside the circle; the tangents are equal. Construction : Draw a line from point P to the center of the circle and join OT & OQ also. To prove : l (PT) = l (PQ) In ? PQO and ? PTO OQ = OT OP = OP PQO = PTO = 90 o Thus , ? PQO ? PTO PT = PQ l (PT) = l (PQ) ? Common side in both triangles ? Right-angle Hypotenuse Side Rule ? (C.S.C.T) Corresponding Side of Congruent Triangle ? Radius of same circle ? Each 90 o Theorem 1 The tangent at any point of a circle is perpendicular to the radius through the point of contact. To prove : OP is perpendicular to xy i.e OP - xy , OQ > OP If we take any point Q on xy other than P , it will be outside the circle, because if Q lies inside the circle, xy will become a secant and not a tangent, as the tangent touches the circle at a single point. Likewise , if we take any Point Q, other than P , it will lie outside the circle. This proves that OP is the shortest distance between point O and Tangent xy . Therefore, OP - to tangent xy T O Q P = = ? ? T O Q P = = ? ? O P x ? ? O P Q Y x ? ? CIRCLES & AREAS RELATED TO CIRCLES 26 Distance covered by a wheel of a car in one revolution is equal to the circumference of the wheel. Total distance covered Circumference of the wheel = ? ? ? ? ? ? When we say ‘segment’ or ‘sector’ we mean the ‘minor segment’ and the ‘minor sector’ . Number of revolutions completed by the wheel The distances between two parallel tangents drawn to a circle is equal to the diameter of the circle O P r r Q If two circles touch internally or externally , the point of contact lies on the straight line through the two centers. P O r Q R { P O r Q R Distance between the centers of two touching circle. R - r r + R PLEASE KEEP IN MIND Areas related to a Circle - Part 1 Amazing tricks to remember circle formula Introduction to circles Tangent to a circle Circles- Theorem - part 1 SCcan the QR Codes to watch our free videos To calculate the shaded/ required area in any gure which is related to circle, rst observe all the gures which are involved in the diagram like square, rectangle, triangle, and circle. And then analyse accordingly, the areas which are to be added or subtracted to get required shaded region. CIRCLES & AREAS RELATED TO CIRCLES 27Read More

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