1 / 5

Explain why the perpendicular from the centre to a chord bisects the chord

Congruent means exactly the same size and shape (reflected shapes can be congruent). ‘Bisects’ means cuts in half (exactly!). Next Circle Theorem. Explain why the perpendicular from the centre to a chord bisects the chord .

salome
Download Presentation

Explain why the perpendicular from the centre to a chord bisects the chord

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Congruent means exactly the same size and shape (reflected shapes can be congruent) ‘Bisects’ means cuts in half (exactly!). Next Circle Theorem Explain why the perpendicular from the centre to a chord bisects the chord To explain why it bisects the chord, you need to show that triangles OBA and OBC are congruent, so you know that the lengths AB and BC are equal. Click for definitions - Bisects - Congruent To show they are congruent you need to show that 2 sides and 1 angle are equal Length 1 Length 2 Angle Answer OB is perpendicular to AC For triangles OBC and OBA: OB is common OA = OC (as they are both radius) Angle OBA = angle OBC (=90°) So triangles OBC and OBA are congruent (identical) O A So AB = BC Hence OB bisects AC. B C

  2. C 40° O 80° A B x x Next Circle Theorem The angle subtended by an arc at the centre of the circle is twice the angle subtended by the arc at any point on the remaining part of the circumference “In other words, the angle at the centre is twice the angle at the circumference” At any point on the circumference! C O 2x A B

  3. 60° 60° 60° A B Next Circle Theorem The angle subtended at the circumference by a semicircle is a right angle “In other words, the angle in a semicircle is always 90°” Anywhere in the circle………… At any point on the circumference……. Continue Angles subtended at the circumference by the same arc are equal C D E ACB = ADB = AEB

  4. Questions Opposite angles of a cyclic quadrilateral sum to 180 degrees In other words, if you draw a quadrilateral (a shape with 4 sides) in a circle then its opposite angles will add up to 180 degrees D Angle A + Angle C = 180° C Angle B + Angle D = 180° B A

  5. A = 51° (Line from centre of circle to midpoint of chord is perpendicular to chord, and angles in triangle) B = 90° (angle in semicircle) C = 48° (angle in a triangle) D = 42° (radius and tangent are perpendicular) E = 27°, F = 27° (angles in same segment are equal) H = 100°, I = 85° (Opposite angles in cyclic quadrilateral J = 100° (angles on a straight line) K = 142°, (triangle containing angle K is isosceles, and angles in a triangle) L = 71° (angle at centre = twice angle at circumference) O = 90° (angle in a semicircle) M = 40° (angles in a triangle) N = 40° (angles in same segment are equal) Questions on Circle Theorems In all questions O is the centre of the circle. Calculate the named angles giving reasons for your answers. Answers (1) (2) (3) E 27° F 42° 39° O O B C A D (4) (5) (6) H 50° 95° L O I N K 19° M 80° J

More Related