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Circle Theorems (Rules and example questions)
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Circle Theorems Rule 1: Rule 2: A D Rute 3: Tangent meets Radius at 90° -B (Tangents that meet at a point are equal in length AB=BC The two rules above B/Cre often seen in a diagram together like this one. AB=BC Angle BAC and BCA = 90° The centre line is a length of both triangles so triangle ABD = BCD. The triangles are identical. Two radi form an Isosceles triangle Rule 4: b a Rule 2 and Rule 3 are Often seen in a diagram together like this one. Rule 5: Rule 6: Rule 7: Rule 8: Opposite Angles in a cyclic Quadrilateral add to 180° a+c=180° b+d=180° Angles at the centre of a circle are' twice the Size of angles at the circumference (arrowhead) Angles in the same segment are equal (bow tie) 4= % The angle at the circumference in a triangle is 90° as long as the triangle is in a semi- circle. Altemate Segment Theorem Angles in alternate segments are equal. A perpendicular line meets a chord at 90° Circle Theorem Questions 1 A, B, C and D are points on a circle. Find the size of angle ACD. Give reasons for your answer. 2 3 Its always the middle angie. ACD=63° as angles in the same segment are equal (bow, tie). 2 In the diagram, A, B, C and D are points on the circle centre O. (a) Work out the size of the angle marked x. Give a reason for your answer. x=148 Angles at the centre are 3 In the diagram, B, D and E are points on the circle centre O. ABC is a tangent to the circle. BE is the diameter of the circle. twice the size of angles at the circumference. (b) Work out the size of...
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Alternative transcript:
the angle marked y. Give a reason for your answer. (a) Work out the size of angle ABD. Give a reason for your answer. Tangent meets radius at 90⁰ 90-36=54° 63° 249 ABD = = 54° 180-74=106° y=106⁰ Opposite angles in a cyclic quadrilateral add to 180° A B ABD 90+36=126 180-126=540 Angles in a triangle add to 180°. D (2 mar 136% (2 marl E (2 mar] (2 mark (b) Work out the size of angle DEB. Give a reason for your answer. Angle B.DE = 90° as the angle at the circumference of a triangle in a semi-circle, is a 90° angle. 3 4 A, B and C are points on the circumference of the circle centre O. The line XCY is the tangent at C to the circle. AB = CB. Work out the size of angle OCB. Give reasons for your answer. Angle ABC - S4° as Alternate Segment Theorem. Triangle ABC is an Isosceles triangle. Angles in a triangle add to 180° 180-54-126 126÷2=63⁰ Angle BAC and Angle BCA are 63° B 86° Diagram NOT accurately drawn A and B are points on the circumference of a circle, centre O. PA and PB are tangents to the circle. Angle APB is 86°. Work out the size of the angle marked x. 3 A tangent meets a radius at 90°. A OAP APO 86÷2=43° as is OPB 90-54-36° 0 PA = PB as tangents that meet at a point are equal in length. B OCA=36° @ OBP and OAP = 90° as tangent meets radius at 90⁰ OCB 63-36=27° OCB=27° is congruent to довр 4 Angles in a triangle add to 180° 90+43=133 180-133=47° (5 marks 5 Angle 0 = 47 1X2 94° 6 A OAB is an Isosceles triangle as formed from two radić Ⓒ180-94 = 86 86÷2=43° x=43° 6. *14. R and S are two points on a circle, centre O. TS is a tangent to the circle. Angle RST = x. Prove that angle ROS = 2x. You must give reasons for each stage of your working. T S and T are points on the circumference of a circle, centre O. PT is a tangent to the circle. SOP is a straight line. Angle OPT=32° Work out the size of the angle marked .x. Give reasons for your answer. Angle PTO radius at 90⁰ 90° Diagram NOT accurately drawn Angle POT =) 90 +32 122 180 -122 32° 58° Angle OST = 90° as radius meets tangent at 90°. OSR=90-x ORS = 90-x as two radii. form an Isosceles triangle. Angles in a triangle add to 180. 90-x+90-x=180-2x 180- (180-2x) 180-180+2x = 2x Diagram NOT accurately drawn P as tangent meets Angles in a Eriangle add to 180° Anglès on a straight add to 180° Angle SOT 180-58 = 122. A OST is an Isosceles triangle as formed from two radii 180-122=58 58÷2=29° x=29°
Circle Theorems (Rules and example questions)
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Shaz
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Circle theorems rules and some example questions that have been gone through.
Shaz
22 Followers
Circle theorems rules and some example questions that have been gone through.
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Circle Theorems Rule 1: Rule 2: A D Rute 3: Tangent meets Radius at 90° -B (Tangents that meet at a point are equal in length AB=BC The two rules above B/Cre often seen in a diagram together like this one. AB=BC Angle BAC and BCA = 90° The centre line is a length of both triangles so triangle ABD = BCD. The triangles are identical. Two radi form an Isosceles triangle Rule 4: b a Rule 2 and Rule 3 are Often seen in a diagram together like this one. Rule 5: Rule 6: Rule 7: Rule 8: Opposite Angles in a cyclic Quadrilateral add to 180° a+c=180° b+d=180° Angles at the centre of a circle are' twice the Size of angles at the circumference (arrowhead) Angles in the same segment are equal (bow tie) 4= % The angle at the circumference in a triangle is 90° as long as the triangle is in a semi- circle. Altemate Segment Theorem Angles in alternate segments are equal. A perpendicular line meets a chord at 90° Circle Theorem Questions 1 A, B, C and D are points on a circle. Find the size of angle ACD. Give reasons for your answer. 2 3 Its always the middle angie. ACD=63° as angles in the same segment are equal (bow, tie). 2 In the diagram, A, B, C and D are points on the circle centre O. (a) Work out the size of the angle marked x. Give a reason for your answer. x=148 Angles at the centre are 3 In the diagram, B, D and E are points on the circle centre O. ABC is a tangent to the circle. BE is the diameter of the circle. twice the size of angles at the circumference. (b) Work out the size of...
Circle Theorems Rule 1: Rule 2: A D Rute 3: Tangent meets Radius at 90° -B (Tangents that meet at a point are equal in length AB=BC The two rules above B/Cre often seen in a diagram together like this one. AB=BC Angle BAC and BCA = 90° The centre line is a length of both triangles so triangle ABD = BCD. The triangles are identical. Two radi form an Isosceles triangle Rule 4: b a Rule 2 and Rule 3 are Often seen in a diagram together like this one. Rule 5: Rule 6: Rule 7: Rule 8: Opposite Angles in a cyclic Quadrilateral add to 180° a+c=180° b+d=180° Angles at the centre of a circle are' twice the Size of angles at the circumference (arrowhead) Angles in the same segment are equal (bow tie) 4= % The angle at the circumference in a triangle is 90° as long as the triangle is in a semi- circle. Altemate Segment Theorem Angles in alternate segments are equal. A perpendicular line meets a chord at 90° Circle Theorem Questions 1 A, B, C and D are points on a circle. Find the size of angle ACD. Give reasons for your answer. 2 3 Its always the middle angie. ACD=63° as angles in the same segment are equal (bow, tie). 2 In the diagram, A, B, C and D are points on the circle centre O. (a) Work out the size of the angle marked x. Give a reason for your answer. x=148 Angles at the centre are 3 In the diagram, B, D and E are points on the circle centre O. ABC is a tangent to the circle. BE is the diameter of the circle. twice the size of angles at the circumference. (b) Work out the size of...
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iOS User
I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D
Stefan S, iOS User
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SuSSan, iOS User
Love this App ❤️, I use it basically all the time whenever I'm studying
Alternative transcript:
the angle marked y. Give a reason for your answer. (a) Work out the size of angle ABD. Give a reason for your answer. Tangent meets radius at 90⁰ 90-36=54° 63° 249 ABD = = 54° 180-74=106° y=106⁰ Opposite angles in a cyclic quadrilateral add to 180° A B ABD 90+36=126 180-126=540 Angles in a triangle add to 180°. D (2 mar 136% (2 marl E (2 mar] (2 mark (b) Work out the size of angle DEB. Give a reason for your answer. Angle B.DE = 90° as the angle at the circumference of a triangle in a semi-circle, is a 90° angle. 3 4 A, B and C are points on the circumference of the circle centre O. The line XCY is the tangent at C to the circle. AB = CB. Work out the size of angle OCB. Give reasons for your answer. Angle ABC - S4° as Alternate Segment Theorem. Triangle ABC is an Isosceles triangle. Angles in a triangle add to 180° 180-54-126 126÷2=63⁰ Angle BAC and Angle BCA are 63° B 86° Diagram NOT accurately drawn A and B are points on the circumference of a circle, centre O. PA and PB are tangents to the circle. Angle APB is 86°. Work out the size of the angle marked x. 3 A tangent meets a radius at 90°. A OAP APO 86÷2=43° as is OPB 90-54-36° 0 PA = PB as tangents that meet at a point are equal in length. B OCA=36° @ OBP and OAP = 90° as tangent meets radius at 90⁰ OCB 63-36=27° OCB=27° is congruent to довр 4 Angles in a triangle add to 180° 90+43=133 180-133=47° (5 marks 5 Angle 0 = 47 1X2 94° 6 A OAB is an Isosceles triangle as formed from two radić Ⓒ180-94 = 86 86÷2=43° x=43° 6. *14. R and S are two points on a circle, centre O. TS is a tangent to the circle. Angle RST = x. Prove that angle ROS = 2x. You must give reasons for each stage of your working. T S and T are points on the circumference of a circle, centre O. PT is a tangent to the circle. SOP is a straight line. Angle OPT=32° Work out the size of the angle marked .x. Give reasons for your answer. Angle PTO radius at 90⁰ 90° Diagram NOT accurately drawn Angle POT =) 90 +32 122 180 -122 32° 58° Angle OST = 90° as radius meets tangent at 90°. OSR=90-x ORS = 90-x as two radii. form an Isosceles triangle. Angles in a triangle add to 180. 90-x+90-x=180-2x 180- (180-2x) 180-180+2x = 2x Diagram NOT accurately drawn P as tangent meets Angles in a Eriangle add to 180° Anglès on a straight add to 180° Angle SOT 180-58 = 122. A OST is an Isosceles triangle as formed from two radii 180-122=58 58÷2=29° x=29°