Circles

Which statement best summarizes the chapter "Circles"?

1. The chapter studies only the area of circles.
2. The chapter explains tangents to circles, the properties of tangents, equal tangent lengths from an external point, and the relationship between the radius and tangent, forming the basis for solving geometric proofs involving circles.
3. The chapter studies only diameters.
4. The chapter has no practical applications.

Which of the following can never be a tangent to a circle?

1. A line touching one point
2. A line cutting the circle at two points
3. A line perpendicular to a radius at the point of contact
4. A line touching only one point

A tangent and a radius at the point of contact form which angle?

1. 30°
2. 45°
3. 60°
4. 90°

Which theorem proves that two tangents from an external point are equal?

1. Pythagoras Theorem
2. Equal Tangent Theorem
3. Midpoint Theorem
4. Basic Proportionality Theorem

The centre of a circle is joined to the point of contact of a tangent. The resulting line segment is:

1. Chord
2. Radius
3. Diameter
4. Secant

If OP is the radius and PT is the tangent at point P, then angle OPT equals:

1. 45°
2. 60°
3. 90°
4. 180°

If the radius of a circle is 7 cm, the diameter is:

1. 7 cm
2. 10 cm
3. 14 cm
4. 21 cm

The point where a tangent touches the circle is called the:

1. Centre
2. Radius
3. Point of Contact
4. Diameter

A circle can have how many tangents?

1. One
2. Two
3. Four
4. Infinitely many

A line that intersects a circle at two distinct points is called:

1. Radius
2. Tangent
3. Secant
4. Diameter

The lengths of two tangents drawn from the same external point to a circle are:

1. Unequal
2. Equal
3. Double each other
4. Zero

How many tangents can be drawn from an external point to a circle?

1. One
2. Two
3. Three
4. Infinite

A tangent to a circle touches the circle at:

1. Two points
2. Three points
3. One point only
4. No point

A chord subtends an angle of 96° at the centre of a circle. What angle will the same chord subtend at any point on the remaining part of the circle?

1. 42°
2. 44°
3. 48°
4. 52°

Tangents PA and PB are drawn from an external point P to a circle. If PA = 5x + 3 and PB = 7x – 9, find the value of x.

1. 5
2. 6
3. 7
4. 8

Two circles have radii of 9 cm and 6 cm. They touch each other externally. What is the distance between their centres?

1. 3 cm
2. 12 cm
3. 15 cm
4. 18 cm

A chord of a circle is 30 cm long. The perpendicular distance from the centre of the circle to the chord is 8 cm. What is the radius of the circle?

1. 15 cm
2. 16 cm
3. 17 cm
4. 18 cm

From an external point P, two tangents PA and PB are drawn to a circle with centre O. If angle APB is 55°, what is the measure of angle AOB?

1. 115°
2. 120°
3. 125°
4. 135°