Circles - Theorems and Properties for Competitive Exams

Circles - Theorems and Properties - Problem Set

Q. If a triangle is inscribed in a circle, what is the relationship between the triangle's angles and the circle's angles?

Solution

The angles of the triangle inscribed in a circle are supplementary to the angles subtended by the same arc at the center.

Correct Answer: B — They are supplementary

Q. If two chords AB and CD intersect at point E inside a circle, and AE = 3 cm, EB = 4 cm, what is the length of segment CE if ED = 6 cm?

Solution

By the intersecting chords theorem, AE * EB = CE * ED. Thus, 3 * 4 = CE * 6, leading to CE = 2 cm.

Correct Answer: C — 7 cm

Q. If two chords AB and CD intersect at point E inside a circle, and AE = 3 cm, EB = 4 cm, what is the length of segment CE if DE = 2 cm?

Solution

By the intersecting chords theorem, AE * EB = CE * DE. Thus, 3 * 4 = CE * 2, which gives CE = 6 cm.

Correct Answer: A — 6 cm

Q. In a circle, if the diameter is 12 cm, what is the circumference?

Solution

Circumference = πd = π(12) = 12π cm.

Correct Answer: B — 24π

Q. What is the length of the arc of a circle with a radius of 7 cm that subtends an angle of 60 degrees at the center?

Solution

Arc length = (θ/360) * 2πr = (60/360) * 2π(7) = (1/6) * 14π = 7π/3 cm.

Correct Answer: A — 7π/3

Q. What is the radius of a circle if its area is 50π square units?

Solution

Area = πr², so 50π = πr². Dividing both sides by π gives r² = 50, thus r = √50 = 5√2.

Correct Answer: A — 5

Q. What is the relationship between the angles formed by two tangents drawn from a point outside a circle?

Solution

The angles formed by two tangents drawn from a point outside a circle to the points of tangency are equal.

Correct Answer: A — They are equal

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