Circle Theorems Mixed Demonstration This resource provides animated

Circle Theorems – Mixed – Demonstration This resource provides animated demonstrations of the mathematical method. Check animations and delete slides not needed for your class.

The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc. ① 180° in a triangle. 360° in a full turn. Base angles in isosceles triangles are equal.

① The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc.

Every angle subtended at the circumference of a semicircle by the diameter is a right angle. ② 180° in a triangle. 360° in a full turn. Base angles in isosceles triangles are equal.

② Every angle subtended at the circumference of a semicircle by the diameter is a right angle.

③ ① Angles subtended at the circumference in the same segment of a circle are equal. The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc. ∴ Angles – in the same segment – created from the same points are equal.

③ Angles subtended at the circumference in the same segment of a circle are equal.

④ ① The sum of the opposite angles of a cyclic quadrilateral is 180°. The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc.

④ The sum of the opposite angles of a cyclic quadrilateral is 180°.

⑤ A tangent to a circle is perpendicular to the radius drawn to the point of contact. A vertical line has 2 points of contact. One point of contact.

⑤ A tangent to a circle is perpendicular to the radius drawn to the point of contact.

Tangents to a circle from an external point to the points of contact are equal in length. ⑥ OA = OB Both are a radius. Both are right-angled triangles. OT is a common side. A ∴Triangles AOT & BOT are congruent (RHS). O T So, AT = BT B

⑦ The line joining an external point to the centre of the circle bisects the angle between the tangents. OA = OB Both are a radius. Both are right-angled triangles. OT is a common side. A ∴Triangles AOT & BOT are congruent (RHS). T So, O AT = BT ∠ATO = ∠BTO B

⑥ Tangents to a circle from an external point to the points of contact are equal in length. ⑦ The line joining an external point to the centre of the circle bisects the angle between the tangents. A Triangles AOT & BOT are congruent (RHS). T So, O AT = BT ∠ATO = ∠BTO B

⑧ A radius bisects a chord at 90°. The segment is bisected (cut in two) by a radius. A AP = BP OA = OB Both are a radius. OP is a common side. ∴Triangles AOP & BOP are congruent (SSS). So, P ∠APO = ∠BPO B O

⑧ A radius bisects a chord at 90°. The segment is bisected (cut in two) by a radius. A AP = BP OA = OB Both are a radius. OP is a common side. ∴Triangles AOP & BOP are congruent (SSS). So, P ∠APO = ∠BPO B O

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. ⑨ Angles subtended at the circumference of a semicircle = 90°. Angles in a triangle total 180°. Angle at a tangent & radius. O Any angle subtended at the circumference in the same segment of a circle are equal. Tangent

⑨ The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Tangent

Circle Theorem Match Up! There are no questions! Match each circle theorem to a diagram. Then, sketch the diagram and add any details you want (angles, centre-points, more lines!) to create a question. A) The sum of the opposite angles of a cyclic quadrilateral is 180°. B) Tangents to a circle from an external point to the points of contact are equal in length. C) A radius bisects a chord at 90°. D) The line joining an external point to the centre of the circle bisects the angle between the tangents. E) The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc. F) Angles subtended at the circumference in the same segment of a circle are equal. G) Every angle subtended at the circumference of a semicircle by the diameter is a right angle. H) The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. I) A tangent to a circle is perpendicular to the radius drawn to the point of contact.

Circle Theorem Match Up! D There are no questions! Match each circle theorem to a diagram. Then, sketch the diagram and add any details you want (angles, centre-points, more lines!) to create a question. A) The sum of the opposite angles of a cyclic quadrilateral is 180°. E B B) Tangents to a circle from an external point to the points of contact are equal in length. H C) A radius bisects a chord at 90°. D) The line joining an external point to the centre of the circle bisects the angle between the tangents. E) The angle at the centre of a circle is twice the angle at the circumference when they are both subtended by the same arc. G I C F) Angles subtended at the circumference in the same segment of a circle are equal. G) Every angle subtended at the circumference of a semicircle by the diameter is a right angle. H) The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. I) A tangent to a circle is perpendicular to the radius drawn to the point of contact. F A

Questions? Comments? Suggestions? …or have you found a mistake!? Any feedback would be appreciated . Please feel free to email: tom@goteachmaths. co. uk