Geometry (Circles Part. 3)

Question 1

Inscribed Angle

Answer 1

The angle subtended at a point on the circle by two given points on a circle.

Question 2

Inscribed Angle Theorem

Answer 2

States that an angle “ψ” inscribed in a circle is half of the Central Angle (Ø) that subtends the same arc of the circle.

I.E. “ψ = Ø/2”

Question 3

What is the measure of Angle (ABC) in degrees?

Answer 3

34°

Question 4

What is the length of Arc (AC)?

Answer 4

16π/5

Question 5

Prove the Inscribed Angle Theorem for Case: A.

Answer 5

1. Spot the Isoceles Triangle and label the two congruent angles as ψ.
2. Label the missing inscribed angle as: (180° - Ø) because an angle of a straight line is equal to 180°.
3. Create equation: ψ + ψ + 180° - Ø = 180°
4. Simplify the equation to receive the theorem: “ψ = Ø/2”

Question 6

Prove the Inscribed Angle Theorem for Case: B.

Answer 6

1. Draw a diameter that goes through the Central Angle and the Inscribed Angle.
2. Split the two sets of angles created as ψ_{1 +} ψ₂ and Ø₁ + Ø₂.
3. Now we have two sets of the same situation in Case: A, so we have the two equations: Ø₁ = 2ψ₁ and Ø₂ = 2ψ₂.
4. We add both equations, substitute (Ø₁ + Ø₂) = Ø and

(ψ_{1 +} ψ₂) = ψ to get “ψ = Ø/2”

Question 7

Prove the Inscribed Angle Theorem for Case: C.

Answer 7

1. Create a diameter and label the parts
2. Because of Case: A, we know that Ø₂ = 2ψ.
3. We combine all the angles created to get:

Ø₂ + Ø = 2ψ₂ + 2ψ

4. We substitue Ø₂ = 2ψ to get:

2ψ + Ø = 2ψ₂ + 2ψ

5. Simplify to get:

Ø = 2ψ₂