1. Parts of a Circle
Introduction
Understanding the different parts of a circle is fundamental to mastering circle theorems and solving related mathematical problems. This section will cover the essential terms and concepts, including chords, arcs, segments, and sectors, as well as secants and their properties.
Key Terms and Definitions
1. Circle: A set of points in a plane that are equidistant from a fixed point called the center.
2. Radius: A line segment from the center of the circle to any point on the circle. All radii of a circle are equal in length.
3. Diameter: A chord that passes through the center of the circle. It is the longest chord of the circle and is equal to twice the radius.
4. Chord: A line segment with both endpoints on the circle. Unlike the diameter, a chord does not necessarily pass through the center.
5. Arc: A part of the circumference of a circle. An arc can be a minor arc (less than \(180^\circ\)) or a major arc (more than \(180^\circ\)).
6. Segment: The region between a chord and the corresponding arc. There are two types of segments:
- Minor Segment: The smaller region created by a chord.
- Major Segment: The larger region created by a chord.
7. Sector: The region between two radii and the corresponding arc. Sectors are also classified as:
- Minor Sector: The smaller area enclosed by two radii and the arc.
- Major Sector: The larger area enclosed by two radii and the arc.
8. Secant: A line that intersects a circle at two points. Unlike a chord, a secant extends beyond the circle.
Secants and Chords
Secants and chords play a crucial role in understanding circle theorems. Here’s a deeper look at their properties and significance:
Properties of Chords:
- The perpendicular bisector of a chord passes through the center of the circle.
- Equal chords are equidistant from the center of the circle.
- Chords equidistant from the center of a circle are equal in length.
Formation of Isosceles Triangles:
- Any two radii of a circle form an isosceles triangle with a chord.
- In an isosceles triangle formed by two radii and a chord, the angles opposite the equal sides (radii) are equal.
Right-Angled Triangles:
- When a radius is perpendicular to a chord, it bisects the chord.
- This creates two congruent right-angled triangles within the circle.
2. Arc Lengths and Sector Calculations
Introduction
Understanding how to calculate arc lengths and sector areas is essential for solving problems related to circles. This section will guide you through the methods to find the length of an arc, the angle in a sector, the perimeter of a sector, and the area of a sector. These calculations are foundational for more advanced topics in circle geometry.
Key Concepts
1. Arc Length
The arc length of a circle is a portion of the circumference. The formula for calculating the length of an arc is:
\[
\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r
\]
where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
2. Sector Area
The area of a sector is a portion of the area of the circle. The formula for calculating the area of a sector is:
\[
\text{Sector Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]
where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
3. Perimeter of a Sector
The perimeter of a sector includes the lengths of the two radii and the arc. The formula for calculating the perimeter of a sector is:
\[
\text{Perimeter of Sector} = 2r + \text{Arc Length}
\]
4. Central Angle in a Sector
The central angle of a sector is the angle subtended by the arc at the center of the circle.
It is often given or needs to be calculated based on other given information.
Step-by-Step Examples
Calculate the length of an arc with a central angle of \(60^\circ\) in a circle with a radius of 10 cm.
\[ \text{Arc Length} = \frac{60^\circ}{360^\circ} \times 2\pi \times 10 = \frac{1}{6} \times 20\pi \approx 10.47 \text{ cm} \]
Calculate the area of a sector with a central angle of \(90^\circ\) in a circle with a radius of 8 cm.
\[ \text{Sector Area} = \frac{90^\circ}{360^\circ} \times \pi \times 8^2 = \frac{1}{4} \times 64\pi \approx 50.27 \text{ cm}^2 \]
Calculate the perimeter of a sector with a central angle of \(45^\circ\) in a circle with a radius of 5 cm.
\[ \text{Arc Length} = \frac{45^\circ}{360^\circ} \times 2\pi \times 5 = \frac{1}{8} \times 10\pi \approx 3.93 \text{ cm} \] \[ \text{Perimeter of Sector} = 2 \times 5 + 3.93 = 10 + 3.93 = 13.93 \text{ cm} \]
A circle has a radius of 12 cm and an arc length of 15 cm. Find the central angle.
\[ \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi \times 12 \] \[ 15 = \frac{\theta}{360^\circ} \times 24\pi \] \[ \theta = \frac{15 \times 360^\circ}{24\pi} \approx 71.62^\circ \]
3. Tangent Lines
Introduction
Tangent lines are fundamental concepts in circle geometry. A tangent line touches the circle at exactly one point. Understanding the properties and relationships of tangent lines is crucial for solving various geometric problems.
Key Concepts
Tangent Line
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency.
The radius drawn to the point of tangency is perpendicular to the tangent line. This property is useful for proving various geometric results.
Properties of Tangent Lines
Tangent segments drawn from an external point to a circle are equal in length. If two tangents are drawn to a circle from an external point, then:
\[
\text{PA} = \text{PB}
\]
where \( \text{PA} \) and \( \text{PB} \) are the lengths of the tangents from the external point \( P \) to the points of tangency \( A \) and \( B \), respectively.
4. Circle Theorems
Introduction
Circle theorems are a set of important principles in geometry that describe the relationships between angles, radii, chords, tangents, and arcs within and around circles. Understanding these theorems is crucial for solving complex geometric problems involving circles.
Key Circle Theorems
1. The Angle at the Center Theorem
The angle subtended at the center of a circle by an arc is twice the angle subtended at the circumference by the same arc.
\[
\angle AOB = 2 \times \angle ACB
\]
where \(O\) is the center of the circle, \(A\) and \(B\) are points on the circle, and \(C\) is another point on the circle.
2. The Angle in a Semicircle Theorem
The angle subtended by a diameter at the circumference of the circle is a right angle.
\[
\angle ACB = 90^\circ
\]
where \(AB\) is the diameter and \(C\) is any point on the semicircle.
3. Angles in the Same Segment Theorem
Angles in the same segment of a circle are equal.
\[
\angle APB = \angle AQB
\]
where \(P\) and \(Q\) are points on the same segment subtended by the chord \(AB\).
4. The Opposite Angles of a Cyclic Quadrilateral Theorem
The sum of the opposite angles of a cyclic quadrilateral is \(180^\circ\).
\[
\angle A + \angle C = 180^\circ \quad \text{and} \quad \angle B + \angle D = 180^\circ
\]
where \(ABCD\) is a cyclic quadrilateral.
5. The Alternate Segment Theorem
The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
\[
\angle BAT = \angle ACB
\]
where \(T\) is the tangent at point \(A\), \(AB\) is the chord, and \(C\) is a point on the circle in the alternate segment.
6. The Perpendicular from the Center to a Chord Theorem
The perpendicular from the center of a circle to a chord bisects the chord.
\[
\text{If} \; OM \perp AB, \; \text{then} \; AM = MB
\]
where \(O\) is the center, \(M\) is the midpoint of the chord \(AB\).
5. Circle theorem example questions
Given a circle with center \(O\), and points \(A\), \(B\), and \(C\) on the circle such that \(\angle AOB = 120^\circ\), find \(\angle ACB\).
Solution: \[ \angle ACB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 120^\circ = 60^\circ \]
In a circle with diameter \(AB\), point \(C\) lies on the circumference. Prove that \(\angle ACB\) is a right angle.
Solution: Since \(AB\) is the diameter, by the Angle in a Semicircle Theorem: \[ \angle ACB = 90^\circ \]
In a circle, points \(P\), \(Q\), \(A\), and \(B\) lie on the circumference such that \(PA\) and \(QB\) are chords. If \(\angle APB = 50^\circ\), find \(\angle AQB\).
Solution: By the Angles in the Same Segment Theorem: \[ \angle AQB = \angle APB = 50^\circ \]
In cyclic quadrilateral \(ABCD\), if \(\angle A = 80^\circ\) and \(\angle C = 100^\circ\), find \(\angle B\) and \(\angle D\).
Solution: \[ \angle A + \angle C = 180^\circ \implies 80^\circ + 100^\circ = 180^\circ \] Therefore, \[ \angle B + \angle D = 180^\circ \] If \(\angle A = 80^\circ\) and \(\angle C = 100^\circ\), then: \[ \angle B = 180^\circ – \angle D = 180^\circ – 80^\circ = 100^\circ \] \[ \angle D = 180^\circ – \angle B = 180^\circ – 100^\circ = 80^\circ \]
In a circle, tangent \(TP\) at point \(P\) and chord \(PQ\) intersect at \(P\). If \(\angle TPQ = 35^\circ\), find \(\angle PQT\).
Solution: By the Alternate Segment Theorem: \[ \angle TPQ = \angle PQT = 35^\circ \]
In a circle with center \(O\), chord \(AB\) is bisected by line \(OM\). If \(OM = 6 \text{ cm}\) and \(AB = 10 \text{ cm}\), find the radius of the circle.
Solution: Since \(OM\) is perpendicular to \(AB\) and bisects it: \[ AM = MB = \frac{AB}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \] Using the Pythagorean theorem in \(\triangle OMA\): \[ OA^2 = OM^2 + AM^2 = 6^2 + 5^2 = 36 + 25 = 61 \] Therefore, the radius \(OA\) is: \[ OA = \sqrt{61} \text{ cm} \]