Unit 7: Circles Study Guide Notes

Standard Form of Circle Equation

• Center at (0,0), Radius r:
  • Equation: $x^2 + y^2 = r^2$
• Center at (h,k), Radius r:
  • Equation: $(x - h)^2 + (y - k)^2 = r^2$

Example 1: Writing Circle Equation

• Problem: Write the equation of a circle centered at (2, -3) with a radius of 7.
• Solution:
  • $(x - 2)^2 + (y - (-3))^2 = (7)^2$
  • $(x - 2)^2 + (y + 3)^2 = 49$

Completing the Square for Circle Equations

Example 2: Finding Standard Form, Center, and Radius

• Problem: Find the equation in standard form, center, and radius of the circle given by
  $x^2 - 10x + y^2 + 6y = -9$
• Solution:
  • Complete the square for x: $x^2 - 10x + 25$
    • $(\frac{10}{2})^2 = (-5)^2 = 25$
  • Complete the square for y: $y^2 + 6y + 9$
    • $(\frac{6}{2})^2 = (3)^2 = 9$
  • Add 25 and 9 to both sides of the equation:
    $x^2 - 10x + 25 + y^2 + 6y + 9 = -9 + 25 + 9$
  • $(x - 5)^2 + (y + 3)^2 = 25$
  • Center: (5, -3)
  • Radius: $r = \sqrt{25} = 5$

Arc Length (Degrees)

• Formula: $s = \frac{\theta}{360} \cdot 2\pi r$

Sector Area (Degrees)

• Formula: Sector Area $= \frac{\theta}{360} \cdot \pi r^2$

Central Angle

• Definition: An angle whose vertex is at the center of the circle.
• Relationship: The measure of a central angle is equal to the measure of its intercepted arc.
• Ratio: 1:1

Example 3: Finding Arc Measure

• Problem: If $m\angle D = 40^\circ$, find the measure of arc AC.
• Solution:
  • $mArc AC = 40^\circ$

Inscribed Angle

• Definition: An angle whose vertex lies on the circle.
• Relationship: The measure of an inscribed angle is half the measure of its intercepted arc.

Example 4: Finding Angle Measure

• If $mArc = 60$ and $m\angle = 30$

Angle Inscribed in a Semicircle

• Property: An angle inscribed in a semicircle measures 90 degrees.

Example 5: Finding x

• Problem: If $m\angle A = 3x$, find x when angle A is inscribed in a semicircle.
• Solution:
  • $90 = 3x$
  • $x = 30$

Inscribed Angles (Same Arc)

• Property: Inscribed angles that intercept the same arc are congruent.

Example 6

• Which angles are equal?

Inscribed Quadrilaterals

• Property: Opposite angles of an inscribed quadrilateral are supplementary (add up to 180 degrees).
• $E + G = 180$

Example 7: Finding Angle Measure

• Problem: If $m\angle G = 80^\circ$, what is $m\angle E$?
• Solution:
  • $m\angle E = 180 - 80 = 100^\circ$

Parallel Chords

• Property: Parallel chords intercept congruent arcs on the circle.

Example 8

• Problem: If CD || AB, then which arcs are equal?
• Solution:
  • Arc AC = Arc DB

Chord and Tangent Angles

• Property: The measure of an angle formed by a chord and a tangent is half the measure of the intercepted arc.

Example 9

• If arc ACB is 280°, what is m∠1?
  • m∠1 = 80
  • 1/2 ~= 40

Intersecting Chords (Inside the Circle)

• Property: The measure of the angle formed by two chords intersecting inside the circle is half the sum of the intercepted arcs.
• $m\angle = \frac{Arc1 + Arc2}{2}$

Example 10: Finding Angle Measure

• Problem: If arc PQ is 86° and arc RS is 110°, what is $m\angle 1$?
• Solution:
  • $m\angle 1 = \frac{86 + 110}{2} = \frac{196}{2} = 98^\circ$

Exterior Angles (Tangents, Secants)

• Formula:
  • $Angle(out) = \frac{Big Arc - Little Arc}{2}$

Example 11: Finding Angle Value

• Problem: Find the value of x, given arcs of 120° and 32° intercepted by an exterior angle x.
• Solution:
  • $x = \frac{120 - 32}{2} = \frac{88}{2} = 44^\circ$

Secant-Secant Lengths

• Rule: Outside * Whole = Outside * Whole (OW = OW)

Example 12: Finding Length Value

• Problem: Find the value of x.

• Given: One secant has an outside part of 3 and a whole length of 12. The other secant has an outside part of x and a whole length of x+5.

• Solution:

  • $x(x + 5) = 3(12)$

Chord-Chord Lengths

• Rule: Part * Part = Part * Part

Secant-Tangent Lengths

• Rule: Outside * Whole = (Tangent)^2 or $OW = (tan)^2$

Tangent and Radius Angle

• Property: A tangent line is perpendicular to the radius at the point of tangency, forming a 90-degree angle.
• Tangent Tangent Lengths: Tangents that meet at the same exterior point are congruent.
• If CE=48, BE=30, and DE=35. Find the length of AE.
  • AE. 30 X 48 = 35
  • 30x= 1680
  • x= 56

Warm-up Problem: Proving a Relationship in Circle K

• Given: Circle K with points B, C, D, and E on the circle; secants HBD and HCE are drawn.
• Prove: HE · DC = HD · EB

Proof: