Circle Theorems

Overview

Circle theorems describe the properties and relationships of lines, angles, and segments in and around a circle. These theorems are essential for solving problems and proving geometric relationships involving circles.

Intersecting Chords Theorem

Statement: If two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
Why It Works: This arises from the similarity of triangles formed by the intersecting chords.

Angles in a Semicircle

Statement: The angle subtended by a diameter at the circumference of a circle is always a right angle ( $90^\circ$ ).
Why It Works: The diameter subtends a semicircle, and the angle formed at the circumference completes the triangle with two radii. The result is a right triangle.

Cyclic Quadrilateral Theorem

Statement: Opposite angles of a cyclic quadrilateral (a quadrilateral inscribed in a circle) are supplementary.
Why It Works: The opposite angles subtend arcs that together form the full circle ( $360^\circ$ ), leading to their sum being $180^\circ$

Tangent to a Circle

Statement: A tangent to a circle is perpendicular to the radius drawn to the point of contact.
Why It Works: A tangent touches the circle at exactly one point, forming a right angle with the radius at that point.

Alternate Segment Theorem

Statement: The angle between the tangent and a chord drawn at the point of contact is equal to the angle in the alternate segment of the circle.
Why It Works: This results from the angles subtended by the chord in different segments being equal.

Circle Theorems Overview Diagram

Worked Examples

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Example 1: Intersecting Chords Theorem

Problem: Two chords $AB$ and $CD$ intersect at $P$ inside a circle.

If $AP = 4$ , $PB = 6$ , and $CP = 3$ , find $PD$ .

Solution:

Step 1: Apply the Intersecting Chords Theorem:

AP \times PB = CP \times PD

Step 2: Substitute the known values:

4 \times 6 = 3 \times PD

Step 3: Solve for $PD$

PD = \frac{24}{3} = 8

Answer: $PD = 8$

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Example 2: Angles in a Semicircle

Problem: A triangle is inscribed in a circle with one side as the diameter. Prove that the triangle is a right triangle.

Solution:

Step 1: The side as the diameter subtends an angle at the circumference.

Step 2: By the Angles in a Semicircle Theorem, this angle is $90^\circ$

Answer: The triangle is a right triangle.

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Example 3: Cyclic Quadrilateral

Problem: In a cyclic quadrilateral, one pair of opposite angles measures $110^\circ$ and $x$ . Find $x$ .

Solution:

Step 1: Opposite angles in a cyclic quadrilateral are supplementary:

110^\circ + x = 180^\circ

Step 2: Solve for $x$ :

x = 180^\circ - 110^\circ = 70^\circ

Answer: $x = 70^\circ$ .

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Example 4: Tangent to a Circle

Problem: A radius $OA$ is drawn to point $A$ where a tangent meets the circle.

Prove that $\angle OAB = 90^\circ$

Solution:

Step 1: Tangent to a Circle Theorem

According to the theorem, the tangent at a point of contact is perpendicular to the radius drawn to that point.

Step 2: Hence:

\angle OAB = 90^\circ

Answer: $\angle OAB = 90^\circ$

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Example 5: Alternate Segment Theorem

Problem: A tangent touches a circle at point $P$ . A chord $PQ$ is drawn, and the angle between the tangent and the chord is $50^\circ$ .

Find the angle in the alternate segment subtended by $PQ$ .

Solution:

Step 1: By the Alternate Segment Theorem, the angle between the tangent and the chord is equal to the angle in the alternate segment.

Step 2: Hence, the angle in the alternate segment is $50^\circ$

Answer: The angle in the alternate segment is $50^\circ$

Summary

Intersecting Chords Theorem: The products of the lengths of the segments of intersecting chords are equal.
Angles in a Semicircle: The angle subtended by a diameter at the circumference is $90^\circ$
Cyclic Quadrilateral: Opposite angles are supplementary.
Tangent to a Circle: A tangent is perpendicular to the radius at the point of contact.
Alternate Segment Theorem: The angle between a tangent and a chord equals the angle in the alternate segment. These theorems provide powerful tools for solving problems involving circles and their properties.

Circle Theorems Simplified Revision Notes for Leaving Cert Mathematics

Circle Theorems

Overview

Intersecting Chords Theorem

Angles in a Semicircle

Cyclic Quadrilateral Theorem

Tangent to a Circle

Alternate Segment Theorem

Circle Theorems Overview Diagram

Worked Examples

Example 1: Intersecting Chords Theorem

Example 2: Angles in a Semicircle

Example 3: Cyclic Quadrilateral

Example 4: Tangent to a Circle

Example 5: Alternate Segment Theorem

Summary

500K+ Students Use These Powerful Tools to Master Circle Theorems For their Leaving Cert Exams.

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