Question

1. which theorem states that any angle inscribed in a semi - circle is a right angle?

a. thales theorem
b. inscribed angle theorem
c. tangent - radius theorem
d. perpendicular bisector theorem

1. which of the following is a valid method for calculating the sector area?

a. sector area = 0.5×radius²×central angle (in radians)
b. sector area = 2×π×radius
c. sector area = π×radius²
d. sector area = radius×central angle (in radians)

1. if an inscribed angle intercepts an arc of 80°, what is the measure of the inscribed angle?

a. 80°
b. 80°
c. 20°
d. 40°

Explanation:

1. Thales' Theorem states that any angle inscribed in a semi - circle is a right angle. The Inscribed Angle Theorem relates the measure of an inscribed angle to its intercepted arc, the Tangent - Radius Theorem is about the perpendicularity of a tangent and a radius, and the Perpendicular Bisector Theorem is about properties of perpendicular bisectors.
2. The formula for the area of a sector of a circle is $A = 0.5\times r^{2}\times\theta$ (where $r$ is the radius and $\theta$ is the central angle in radians). The other formulas provided are incorrect.
3. The measure of an inscribed angle is half the measure of the intercepted arc. If the intercepted arc is $80^{\circ}$, then the inscribed angle is $\frac{80^{\circ}}{2}=40^{\circ}$.

Answer:

1. a. Thales' Theorem
2. a. Sector Area = 0.5×Radius²×Central Angle (in radians)
3. d. 40°