Circle Geometry (9)

Question 1

What is a tangent to a circle?

Answer 1

A line that touches the circle at exactly one point

Question 2

How does a tangent relate to the radius at the point of contact?

Answer 2

Tangent is perpendicular to the radius at the point of contact (90°)

Question 3

Two tangents are drawn to a circle from the same external point. What is true about their lengths?

Answer 3

Tangents from the same external point are equal in length

Question 4

What is a chord in a circle?

Answer 4

A line segment with both endpoints on the circle

Question 5

What is a diameter of a circle?

Answer 5

A chord that passes through the center; it is the longest chord

Question 6

What is the relationship between a central angle and the angle at the circumference that subtends the same arc?

Answer 6

The central angle is twice the angle at the circumference

Question 7

A triangle is inscribed in a circle and one side is the diameter. What kind of triangle is it?

Answer 7

Right triangle; the angle opposite the diameter is 90°

Question 8

Angles in the same segment of a circle are equal. Explain with an example.

Answer 8

If two angles subtend the same chord/arc, they are equal; e.g., angles on the same side of a chord are equal

Question 9

How do you find the length of an arc if the radius is r and the central angle is θ in degrees?

Answer 9

Arc length = (θ/360) * 2πr

Question 10

How do you find the area of a sector with radius r and central angle θ in degrees?

Answer 10

Sector area = (θ/360) * πr²

Question 11

A circle has radius 6 cm. Find the length of an arc with central angle 120°.

Answer 11

Arc length = (120/360) * 2π*6 = (1/3) * 12π = 4π ≈ 12.57 cm

Question 12

A circle has radius 10 cm. Find the area of a sector with central angle 60°.

Answer 12

Area = (60/360) * π10² = (1/6)100π ≈ 52.36 cm²

Question 13

A tangent meets a circle at point A. A radius from the center meets the circle at A. Draw the angle formed and state its measure.

Answer 13

Right angle (90°) between the radius and tangent

Question 14

An angle at the circumference subtends a 100° arc. What is the central angle of the same arc?

Answer 14

Central angle = 2 * 100° = 200° (if reflex, otherwise angle = 100°? check context; usually central = double the inscribed angle)