Mastering Circle Theorems

What is the angle subtended by a diameter at the circumference?

• 45 degrees

• 90 degrees

• 180 degrees

• 120 degrees

State the chord-chord theorem and provide an example.

• If AB = 7 and AE = 2, then EB = 5. If CD = 9 and CE = 4, then ED = 5. Therefore, 2 * 5 = 4 * 5, which confirms the theorem.

• If AB = 6 and AE = 3, then EB = 3. If CD = 8 and CE = 4, then ED = 4. Therefore, 6 * 3 = 4 * 4, which confirms the theorem.

• If AB = 5 and AE = 1, then EB = 4. If CD = 10 and CE = 2, then ED = 8. Therefore, 1 * 4 = 2 * 8, which confirms the theorem.

• If AB = 6 and AE = 2, then EB = 4. If CD = 8 and CE = 3, then ED = 5. Therefore, 2 * 4 = 3 * 5, which confirms the theorem.

If two chords intersect inside a circle, how do you calculate the angle formed?

• Angle = arc1 + arc2

• Angle = 1/2 * (arc1 + arc2)

• Angle = 1/2 * (arc1 - arc2)

• Angle = arc1 * arc2

Explain the secant-secant theorem with a diagram.

• If AB and CD are secants intersecting at the center of the circle, then (AP * PB) = (CP + PD).

• If AB and CD are secants intersecting outside the circle at P, then (AP * PB) = (CP * PD).

• If AB and CD are tangents intersecting outside the circle at P, then (AP * PB) = (CP / PD).

• If AB and CD are chords intersecting inside the circle at P, then (AP + PB) = (CP + PD).

How do you find the length of a tangent from a point outside the circle?

• length = 2r

• length = √(d² - r²)

• length = d + r

• length = d - r

What is the relationship between the angles formed by two secants intersecting outside a circle?

• The angle is half the sum of the intercepted arcs.

• The angle is half the difference of the intercepted arcs.

• The angle is equal to the difference of the intercepted arcs.

• The angle is equal to the sum of the intercepted arcs.

Describe the tangent-tangent theorem and its application.

• Two tangents from a point inside the circle are equal in length.

• The lengths of the two tangents from an external point to a circle are equal.

• The lengths of the two tangents from an external point to a circle are different.

• The tangent to a circle is always longer than the radius.

If two tangents are drawn from a point outside the circle, what can be said about their lengths?

• One tangent is longer than the other.

• The lengths of the two tangents are equal.

• The lengths of the tangents are proportional to the radius.

• The lengths of the two tangents are different.

How does the angle formed by a tangent and a chord relate to the angle subtended by the chord at the circumference?

• The angle formed by a tangent and a chord is equal to twice the angle subtended by the chord at the circumference.

• The angle formed by a tangent and a chord is complementary to the angle subtended by the chord at the circumference.

• The angle formed by a tangent and a chord is equal to the angle subtended by the chord at the circumference.

• The angle formed by a tangent and a chord is half the angle subtended by the chord at the circumference.

Provide a real-world example where circle theorems are applied.

• The layout of a rectangular garden.

• The calculation of the area of a triangle.

• The design of a square swimming pool.

• The design of a circular dome in architecture.