Aim 15: What relationship exists if a diameter is ...



Aim 15: What relationship exists if a diameter is perpendicular to a chord?

Do Now:

Examine the given measurements, and complete the theorem in the box.

If a diameter is perpendicular to a chord, then it _________ the chord and its major and minor arcs.

True or False: If a line through the center of a circle bisects a chord, then it is also perpendicular to that chord. (Justify your answer.)

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1. In circle E, chord [pic]. 2. In circle A, chord [pic].

If = [pic], find If [pic],[pic], &[pic] find [pic].

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3. Given circle A, [pic] at Y.

a) prove [pic] b) prove [pic]

Staements Reasons

1. Circle A, [pic] at Y. 1. given

Prove: In the same or in congruent circles, if two chords are equidistant from the center, then they are congruent.

1. Given: Circle P with chords [pic].

Since distance is always measured perpendicularly,

[pic] at A, [pic] at I , and [pic].

Prove: [pic]

Statements Reasons

1. Circle A with chords [pic], 1. given

[pic] at A, [pic] at I , & [pic].

2. Radii [pic] are drawn 2. The radius is a line segment from the center of the circle to any point on the circle.

3. [pic] are right angles 3.

4. [pic] 4.

5. [pic] are right triangles 5.

6. [pic] 6.

7. [pic] 7.

8. [pic] 8.

9. [pic] 9.

10. [pic] and [pic] 10.

11. [pic] 11.

12. [pic] 12.

Prove: In a circle, a diameter perpendicular to a chord bisects the chord, and the major and minor arcs.

Given: Circle T with diameter [pic]. Chord [pic] at A.

Prove: 1. [pic] 2. [pic] 3. [pic]

Statements Reasons

1. Circle T with diameter [pic]. 1. given

Chord [pic] at A.

2. Radii [pic] are drawn 2. The radius is a line segment from the center of the circle to any point on the circle.

3. [pic] 3.

4. [pic] are right angles 4.

5. [pic] 5.

6. [pic] is isosceles 6.

7. [pic] 7.

8. [pic] 8.

9. [pic] and 9.

[pic]

10. [pic] 10.

11. [pic] 11. In a circle, the diameter divides it into two congruent semi-circles.

12. - [pic] - 12.

or [pic]

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