Geometry (HS pathway)

Circles: Arcs, Angles, and Equations

Geometry

✓By the end of this lesson students will be able to define and identify central angles, inscribed angles, and various types of arcs.
✓By the end of this lesson students will be able to calculate arc measures and angle measures related to central and inscribed angles using relevant theorems.
✓By the end of this lesson students will be able to write the standard form equation of a circle given its center and radius, or other sufficient information.
✓By the end of this lesson students will be able to identify the center and radius of a circle from its standard form equation and graph the circle.
✓By the end of this lesson students will be able to convert the general form equation of a circle to its standard form by completing the square.

Key concepts

Central Angle and Intercepted Arc

A central angle is an angle whose vertex is the center of the circle and whose sides are radii. The measure of a central angle is equal to the measure of its intercepted arc. An intercepted arc is the portion of the circle that lies in the interior of the angle and has endpoints on the angle's sides.

m(central angle) = m(intercepted arc)

Inscribed Angle and Intercepted Arc

An inscribed angle is an angle whose vertex is on the circle and whose sides are chords of the circle. The measure of an inscribed angle is half the measure of its intercepted arc.

m(inscribed angle) = 1/2 * m(intercepted arc)

Arc Measures

A minor arc is an arc whose measure is less than 180 degrees. Its measure is equal to its central angle. A major arc is an arc whose measure is greater than 180 degrees. Its measure is 360 degrees minus the measure of its related minor arc. A semicircle is an arc whose endpoints are the endpoints of a diameter, and its measure is 180 degrees.

Inscribed Angle Theorems

If two inscribed angles intercept the same arc, then the angles are congruent. An angle inscribed in a semicircle is a right angle (90 degrees).

Standard Form of the Equation of a Circle

The standard form of the equation of a circle defines a circle with a specific center and radius. The center of the circle is represented by the coordinates (h, k), and 'r' represents the radius.

(x - h)^2 + (y - k)^2 = r^2

General Form of the Equation of a Circle

The general form of the equation of a circle is a polynomial equation where the coefficients of x^2 and y^2 are equal. To find the center and radius from this form, you must convert it to standard form by completing the square.

Ax^2 + Ay^2 + Bx + Cy + D = 0 (where A ≠ 0)

Key facts to remember

1The measure of a central angle is equal to the measure of its intercepted arc.
2The measure of an inscribed angle is half the measure of its intercepted arc.
3An angle inscribed in a semicircle is always a right angle (90 degrees).
4The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
5To convert a circle's equation from general form to standard form, use the method of completing the square.
6The distance formula can be used to find the radius if the center and a point on the circle are known.
7The midpoint formula can be used to find the center of a circle if the endpoints of a diameter are known.

Worked examples

Example 1

In a circle with center O, points A, B, C are on the circumference. If the measure of central angle AOB is 80 degrees, find the measure of arc AB and the measure of inscribed angle ACB.

IStep 1: Identify the relationship between the central angle and its intercepted arc. The measure of a central angle is equal to the measure of its intercepted arc.

IIm(arc AB) = m(angle AOB)

IIIm(arc AB) = 80 degrees

IVStep 2: Identify the relationship between the inscribed angle and its intercepted arc. The measure of an inscribed angle is half the measure of its intercepted arc.

Vm(angle ACB) = 1/2 * m(arc AB)

VIm(angle ACB) = 1/2 * 80 degrees

VIIm(angle ACB) = 40 degrees

Answer

The measure of arc AB is 80 degrees. The measure of angle ACB is 40 degrees.

Always distinguish between central angles and inscribed angles when determining arc measures.

Example 2

Write the standard form equation of a circle with center (-4, 7) and a radius of 6.

IStep 1: Identify the coordinates of the center (h, k) and the radius r.

IIh = -4, k = 7, r = 6

IIIStep 2: Substitute these values into the standard form equation of a circle: (x - h)^2 + (y - k)^2 = r^2.

IV(x - (-4))^2 + (y - 7)^2 = 6^2

VStep 3: Simplify the equation.

VI(x + 4)^2 + (y - 7)^2 = 36

Answer

(x + 4)^2 + (y - 7)^2 = 36

Be careful with the signs when substituting h and k into the equation.

Example 3

Find the center and radius of the circle given by the equation x^2 + y^2 + 10x - 6y - 2 = 0.

IStep 1: Group the x-terms and y-terms, and move the constant term to the right side of the equation.

II(x^2 + 10x) + (y^2 - 6y) = 2

IIIStep 2: Complete the square for the x-terms and the y-terms. To do this, take half of the coefficient of the x-term (10/2 = 5) and square it (5^2 = 25). Do the same for the y-term (-6/2 = -3, and (-3)^2 = 9). Add these values to both sides of the equation.

IV(x^2 + 10x + 25) + (y^2 - 6y + 9) = 2 + 25 + 9

VStep 3: Factor the perfect square trinomials and simplify the right side.

VI(x + 5)^2 + (y - 3)^2 = 36

VIIStep 4: Compare this equation to the standard form (x - h)^2 + (y - k)^2 = r^2 to identify the center (h, k) and the radius r.

VIIIh = -5, k = 3, r^2 = 36, so r = sqrt(36) = 6

Answer

The center of the circle is (-5, 3) and the radius is 6.

Remember to add the values used to complete the square to BOTH sides of the equation to maintain equality.

Common mistakes

✗Confusing the relationship between central angles (equal to arc) and inscribed angles (half the arc).
✗Incorrectly identifying the center (h, k) from the standard form equation, especially sign errors (e.g., (x+3)^2 means h = -3, not 3).
✗Forgetting to add the values used to complete the square to both sides of the equation when converting from general to standard form.
✗Calculating r^2 instead of r for the radius from the standard form equation.
✗Assuming any chord is a diameter without verifying it passes through the center.

Exam tips

★Always draw a clear diagram for problems involving arcs and angles to visualize the relationships.
★Memorize the standard form of the circle equation and the key theorems for central and inscribed angles.
★Pay close attention to positive and negative signs when identifying the center coordinates (h, k) from the equation.
★Practice completing the square regularly, as it is a fundamental skill for circle equations in general form.
★Double-check your calculations, especially when dealing with squares and square roots for the radius.

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