Circles - Theorems and Properties for Exam Success

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Circles - Theorems and Properties - Proof-based Questions - Applications

Q. If a tangent from point P touches a circle at point T, and the radius OT is 5 cm, what is the length of PT if PT is perpendicular to OT?

A. 5√2 cm
B. 5 cm
C. 10 cm
D. 25 cm

Solution

Since PT is perpendicular to OT, PT is equal to the radius OT, which is 5 cm.

Correct Answer: B — 5 cm

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Q. If two chords AB and CD of a circle intersect at point E, and AE = 3 cm, EB = 5 cm, what is the length of segment CE if ED = 4 cm?

A. 6 cm
B. 8 cm
C. 12 cm
D. 10 cm

Solution

According to the intersecting chords theorem, AE * EB = CE * ED. Thus, 3 * 5 = CE * 4, which gives CE = (15/4) = 3.75 cm.

Correct Answer: B — 8 cm

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Q. In a circle, if the length of an arc is 15 cm and the radius is 10 cm, what is the angle in radians subtended by the arc at the center?

A. 1.5 radians
B. 0.5 radians
C. 2 radians
D. 3 radians

Solution

The angle θ in radians is given by the formula θ = arc length / radius. Therefore, θ = 15/10 = 1.5 radians.

Correct Answer: A — 1.5 radians

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Q. In a circle, if the radius is 10 cm, what is the area of the circle?

A. 100π cm²
B. 50π cm²
C. 25π cm²
D. 200π cm²

Solution

The area of a circle is given by the formula A = πr². Therefore, A = π(10)² = 100π cm².

Correct Answer: A — 100π cm²

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Q. What is the length of the diameter of a circle if its circumference is 31.4 cm? (Use π = 3.14)

A. 10 cm
B. 5 cm
C. 15 cm
D. 20 cm

Solution

The circumference C of a circle is given by C = πd. Therefore, d = C/π = 31.4/3.14 = 10 cm.

Correct Answer: A — 10 cm

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