Question

Part AConstruct a circle of any radius, and draw a chord on it. Then construct the radius of the circle that bisects the chord. Measure the angle between the chord and the radius. What can you conclude about the intersection of a chord and the radius that bisects it? Take a screenshot of your construction, save it, and insert the image below your answer.Part BWrite a paragraph proof of your conclusion in part A. To begin your proof, draw radii OA and OC.Part CIn this part of the activity, you will investigate the converse of the theorem stated in part A. To get started, reopen GeoGebra.

Answer 1

Given:

Part A

We will construct a circle of any radius, and draw a chord on it. Then construct the radius of the circle that bisects the chord.

The graph of the circle and other construction will be as shown in the following figure.

What can you conclude about the intersection of a chord and the radius that bisects it?

We can conclude that the chord and radius are perpendicular

The measure of the angle OBA = 90°

Part B

Write a paragraph proof of your conclusion in part A. To begin your proof, draw radii OA and OC.

The proof will be as follows:

Statement Reason

1. OA = OC Both are the radius of the circle O

2. AB = CB OB bisects the chord AC

3. OB = OB Reflexive property

4. ΔOBA ≅ Δ OBC By SSS

5. ∠OBA = ∠OBC CPCTC

6. ∠OBA + ∠OBC = 180 Straight angle definition

7. 2 ∠OBA = 180 Substitute from (5)

8. ∠OBA = 90 Simplify (7)

9. OB ⊥ AC Right-angle definition

Part C

The converse of the theorem will be as follows:

The perpendicular from the center of a circle to a chord bisects the chord