This chapter delves into the world of circles, exploring their fundamental properties, how to represent them with equations, and how to find lines tangent to them.
Chapter 10 – Circles
Basic Properties of Circles
A circle is a collection of points equidistant from a fixed point (centre).
- Center (O): The fixed point at the center of the circle.
- Radius (r): The distance from the center (O) to any point on the circle’s circumference. All radii of a circle are equal in length.
- Diameter (d): A straight line segment passing through the center and ending on opposite points of the circle’s circumference. The diameter is equal to twice the radius (d = 2r).
- Circumference (c): The total length of the circle’s boundary. (c = 2πr, where π is a constant approximately equal to 22/7)
Formulas:
- Circumference (c) = 2πr (π is a mathematical constant approximately equal to 3.14159)
- Area (A) = πr²
Equation of a Circle
Circles can be represented by an equation that defines the relationship between the coordinates of any point on the circle and the center’s coordinates.
Formula: (x – h)² + (y – k)² = r²
- (h, k) represent the coordinates of the circle’s center.
- (x, y) represent the coordinates of any point on the circle’s circumference.
- r represents the circle’s radius.
Note: This equation essentially states that the distance between any point on the circle’s circumference and the center is equal to the radius.
Tangents to a Circle
A tangent to a circle is a straight line that touches the circle at exactly one point. The point of contact is called the point of tangency.
Properties of Tangents:
- The radius drawn from the center of the circle to the point of tangency is always perpendicular to the tangent line.
- From an external point, two tangents can be drawn to the circle. These tangents are equal in length.
Examples
1. Finding the Area of a Circle
A circle has a radius of 5 cm. Find the area of the circle.
Solution:
- Use the formula for area: A = πr²
- Substitute the known value of r (5 cm): A = π * (5 cm)²
- Calculate the area: A ≈ 78.54 cm² (using a calculator for π)
2. Writing the Equation of a Circle
A circle is centered at point (2, 3) and has a radius of 4 units. Write the equation of the circle.
Solution:
- Use the general formula for the equation of a circle: (x – h)² + (y – k)² = r²
- Substitute the known values: (x – 2)² + (y – 3)² = 4²
Therefore, the equation of the circle is (x – 2)² + (y – 3)² = 16.
3. Determining Tangent Line Equation
A circle is centered at (0, 0) with a radius of 6 units. A point P lies outside the circle at (3, 4). Find the equation of the tangent line drawn from point P to the circle.
Solution:
- Slope of the Tangent:
- Draw a line segment connecting the center (0, 0) and point P (3, 4). This line segment is the radius since it connects the center to a point on the circle.
- Calculate the slope of this line segment (radius) using the slope formula: m = (y2 – y1) / (x2 – x1) = (4 – 0) / (3 – 0) = 4/3
- Perpendicular Slope for Tangent:
- The tangent line will have a slope that is negative reciprocal of the radius’s slope. So, the slope of the tangent line (m_tangent) is -3/4.
- Point-Slope Form for Tangent:
- We know the slope of the tangent line (-3/4) and the coordinates of the external point P (3, 4).
- Use the point-slope form for the tangent line equation: y – y1 = m_tangent(x – x1)
- Substitute the known values: y – 4 = -3/4(x – 3)
Therefore, the equation of the tangent line is y – 4 = -3/4(x – 3).
Arcs
- An arc is a portion of a circle’s circumference. It can be defined by its two endpoints on the circle.
- Types of arcs:
- Minor arc: The shorter arc between the two endpoints on the circle.
- Major arc: The longer arc between the two endpoints on the circle.
- Central angle: The angle formed at the center of the circle by the two radii drawn to the endpoints of the arc.
- Measure of an arc: The measure of an arc is equal to the ratio of the central angle’s measure to 360° (or 2π radians) multiplied by the circle’s circumference.
Chords and Secants
- Chord: A line segment that connects two points on a circle’s circumference.
- Secant: A line that intersects the circle at two distinct points.
- Properties:
- A diameter is a special type of chord that passes through the center of the circle and divides it into two equal halves.
- The perpendicular bisector of a chord passes through the center of the circle. (This will be proven in more advanced courses)
- If two secants intersect outside the circle, the product of the lengths of one secant’s two segments is equal to the product of the lengths of the other secant’s two segments (Power of a Point Theorem – introduced in more advanced courses).
Examples
1. Finding the Measure of a Minor Arc
A circle has a radius of 8 cm. A minor arc on this circle subtends a central angle of 120°. Find the measure of the minor arc.
Solution:
- Calculate the circle’s circumference: c = 2πr = 2π * 8 cm ≈ 50.27 cm
- Find the ratio between the central angle and 360°: 120° / 360° = 1/3
- Multiply this ratio by the circle’s circumference to find the arc measure: (1/3) * 50.27 cm ≈ 16.76 cm
Therefore, the measure of the minor arc is approximately 16.76 cm.
2. Determining Length of a Chord
In the same circle from example 1 (radius 8 cm), a chord is drawn that is not a diameter. How can we find the length of the chord without directly measuring it on the circle?
Solution:
- Advanced Approach (Pythagorean Theorem):
- If we know the distance between the center of the circle and the midpoint of the chord (which is also perpendicular to the chord), we can use the Pythagorean theorem along with the radius to solve for the chord length.
- Basic Approach (Applications in later chapters):
- By introducing concepts like inscribed angles and their relationship with arc measures, we can develop formulas to find chord lengths based on the central angle and circle properties (explored in later chapters).
3. Identifying Secant Properties:
Imagine two secants intersecting outside a circle. One secant intersects the circle at points A and B, with segment lengths AC = 6 cm and CB = 8 cm. The other secant intersects at points D and E, with segment lengths DE = 12 cm and ED = x cm (unknown).
Solution:
- Apply the Power of a Point Theorem (which will be proven later): AC * CB = DE * ED
- Substitute the known values: 6 cm * 8 cm = 12 cm * x cm
- Solve for x: x = (6 cm * 8 cm) / 12 cm ≈ 4 cm
Therefore, segment ED on the second secant has a length of approximately 4 cm.
Key Concepts
This chapter dives into circles, exploring their properties, how to represent them with equations, and how to find lines tangent to them. Here’s a quick summary of essential concepts:
- Basic Properties: Center, radius, diameter, circumference (c = 2πr), and area (A = πr²).
- Equation of a Circle: (x – h)² + (y – k)² = r² (h, k represent center; r represents radius).
Short Notes:
- All radii in a circle are equal.
- Diameter is twice the radius (d = 2r).
- This circle equation ensures any point on the circle is a fixed distance (radius) from the center.
Tangents to Circles
- A tangent line touches the circle at exactly one point (point of tangency).
- The radius drawn to the point of tangency is perpendicular to the tangent line.
- Two tangents can be drawn from an external point, and they are equal in length.
Examples (Abbreviated)
- Finding Circle Area: Use A = πr² (given radius).
- Writing Circle Equation: Substitute center and radius values in the general formula.
- Tangent Line Equation: Find the slope of the tangent (negative reciprocal of radius slope) and use the point-slope form with the external point’s coordinates.
Beyond the Basics
- Arcs: Portions of the circle’s circumference.
- Minor arc: Shorter arc between two endpoints.
- Major arc: Longer arc between two endpoints.
- Central angle: Angle formed at the center by radii drawn to the arc’s endpoints.
- Arc measure: Ratio of central angle measure to 360° (or 2π radians) multiplied by circle’s circumference.
- Chords and Secants: Line segments interacting with circles.
- Chord: Connects two points on the circle’s circumference.
- Secant: Intersects the circle at two distinct points.
- Diameter: A special chord passing through the center, dividing the circle into halves.
- Perpendicular bisector of a chord passes through the circle’s center (proven in advanced courses).
- Power of a Point Theorem (introduced later): Relates lengths of segments created by two secants intersecting outside the circle.
Practice Questions (with Explanations)
- Find the circumference and area of a circle with a radius of 10 cm.
- Explanation: c = 2πr = 2 * (22/7) * 10 = 62.86 cm; A = πr² = (22/7) * 10² = 314.16 cm².
- Write the equation of a circle with center (2, 3) and radius 5 cm.
- Explanation: (x – 2)² + (y – 3)² = 5².
Summary
Understanding circles goes beyond basic properties. Arcs, chords, and secants introduce additional elements with their own properties. While some concepts like the Power of a Point Theorem will be explored further later, this chapter equips you with the tools to analyze circles, find their properties, and solve problems involving their areas, equations, tangents, arcs, chords, and secants. Remember, practice is key to mastering these concepts.
