Honors Geometry: Comprehensive Guide to Circles

Vocabulary and Basic Intersections

• Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center $P$ is titled "circle $P$" and can be expressed notationally as $\bigodot P$.

• Radius: A segment whose endpoints are the center and any point on the circle. In naming, if $C$ is the center and $B$ is on the circle, the radius is $\overline{CB}$.

• Chord: A segment whose endpoints are on a circle.

• Diameter: A chord that specifically contains the center of the circle.

• Secant: A line that intersects a circle in exactly two points.

• \Tangent: A line in the plane of a circle that intersects the circle in exactly one point, known as the point of tangency. This term also applies to tangent rays (e.g., $\vec{AB}$) and tangent segments (e.g., $\overline{AB}$).

• One point of intersection: Known as tangent circles. These can be externally tangent or internally tangent.

• No points of intersection: This occurs when circles are completely separate or when they are concentric circles (coplanar circles that share a common center).

• Common Tangents: A line or segment that is tangent to two coplanar circles.

  • Common Internal Tangent: Intersects the imaginary segment joining the centers of the two circles.

  • Common External Tangent: Does not intersect the segment joining the centers of the two circles.

Tangent Theorems and Geometric Applications

• Theorem 10.1: Tangent Line to Circle Theorem: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. (Line $m \perp \overline{QP}$ iff $m$ is tangent to $\bigodot Q$ at $P$).

• Theorem 10.2: External Tangent Congruence Theorem: Tangent segments from a common external point are congruent. If segments $\overline{SR}$ and $\overline{ST}$ are tangent to the same circle from point $S$, then $SR = ST$.

• Sample Calculation 1 (Finding Radius): For a right triangle formed by a radius $r$, a tangent segment of $\text{12 units}$, and a hypotenuse of $\text{13 units}$:

  • $r^2 + 12^2 = 13^2$

  • $r^2 + 144 = 169$

  • $r^2 = 25$

  • $r = 5$

• Sample Calculation 2 (Verifying Tangency): To determine if a line is tangent, check if the Pythagorean theorem holds ($a^2 + b^2 = c^2$). Example: $37^2 = 35^2 + 12^2 \Rightarrow 1369 = 1225 + 144 \Rightarrow 1369 = 1369$. Result: Yes, it is tangent.

• Sample Calculation 3 (Finding Internal Radius): Given a tangent $\text{24}$, internal radius $r$, and external segment $18$:

  • $r^2 + 24^2 = (r + 18)^2$

  • $r^2 + 576 = r^2 + 36r + 324$

  • $576 = 36r + 324$

  • $252 = 36r$

  • $r = 7$

Measuring Arcs and Central Angles

• Central Angle: An angle whose vertex is the center of the circle and whose sides are radii.

• Circular Arc: A portion of a circle. The measure of an arc is defined by the measure of its central angle.

• Minor Arc: An arc with a central angle less than $180^\circ$. It is named by its two endpoints. Example: $\widehat{AB}$.

• Major Arc: The arc containing all points between two endpoints that are NOT on the minor arc. It has a measure greater than $180^\circ$ and is named by three points. Example: $\widehat{ADB}$.

• Semicircle: An arc with endpoints that are the endpoints of a diameter. Its measure is exactly $180^\circ$.

• Calculating Measures:

  • Measure of a Minor Arc = Measure of Central Angle.

  • Measure of a Major Arc = $360^\circ - \text{Measure of Minor Arc}$.

• Postulate 10.1: Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. $m\widehat{ABC} = m\widehat{AB} + m\widehat{BC}$.

• Geometric Similarity and Congruence:

  • Theorem 10.3: Congruent Circles Theorem: Two circles are congruent iff they have the same radius.

  • Theorem 10.5: Similar Circles Theorem: All circles are similar.

  • Similar Arcs: Arcs that have the same measure. Note: All congruent arcs are similar, but not all similar arcs are congruent (as they may belong to circles with different radii).

Inscribed Angles and Polygons

• Inscribed Angle: An angle whose vertex is ON the circle and whose sides contain chords.

• Intercepted Arc: The arc that lies between the sides of an inscribed angle.

• Theorem 10.10: Measure of an Inscribed Angle: The measure of an inscribed angle is one-half the measure of its intercepted arc.

• Theorem 10.11: Inscribed Angles of a Circle Theorem: If two inscribed angles of a circle intercept the same arc, then the angles are congruent. (e.g., $\angle ADB \cong \angle ACB$).

• Inscribed Polygon: A polygon where all vertices lie on a circle.

• Circumscribed Circle: The circle that contains the vertices of an inscribed polygon.

• Theorem: Right Triangle Inscribed in a Circle: An angle inscribed in a semicircle is a right angle ($90^\circ$).

• Theorem: Inscribed Quadrilateral: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary ($\text{Sum} = 180^\circ$).

  • Example: $\text{Opposite angles } 8x + 10x = 180 \Rightarrow 18x = 180 \Rightarrow x = 10$.

Angle Relationships in Circles

• Theorem 10.14: Tangent and Intersected Chord Theorem: If a tangent and a chord intersect at a point on a circle, the measure of each angle formed is half the measure of its intercepted arc.

• Theorem 10.15: Angles Inside the Circle Theorem: If two chords intersect inside a circle, the measure of each angle is one-half the sum of the measures of the intercepted arcs.

  • $\text{Angle} = \frac{1}{2}(\text{Arc}_1 + \text{Arc}_2)$

• Theorem 10.16: Angles Outside the Circle Theorem: If a tangent and a secant, two tangents, or two secants intersect outside a circle, the measure of the angle formed is half the difference of the intercepted arcs.

  • $\text{Angle} = \frac{1}{2}(\text{Arc}_{\text{far}} - \text{Arc}_{\text{near}})$

• Theorem 10.17: Circumscribed Angle Theorem: The measure of a circumscribed angle (formed by two tangents) is equal to $180^\circ$ minus the measure of the central angle that intercepts the same arc.

Segment Relationships in Circles

• Segments of Chords (Theorem 10.18): 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. $(a \times b = c \times d)$.

  • Example: $4 \times 6 = x \times 3 \Rightarrow 24 = 3x \Rightarrow x = 8$.

• Segments of Secants (Theorem 10.19): If two secant segments share the same endpoint outside a circle, then: $\text{External Segment}_1 \times \text{Whole Secant}_1 = \text{External Segment}_2 \times \text{Whole Secant}_2$.

  • Example: $9(9+11) = 10(10+x) \Rightarrow 180 = 100 + 10x \Rightarrow 80 = 10x \Rightarrow x = 8$.

• Segments of Secants and Tangents (Theorem 10.20): If a secant segment and a tangent segment share an endpoint outside a circle, then: $\text{Tangent}^2 = \text{External Secant} \times \text{Whole Secant}$.

  • Example: $x^2 = 1 \times 4 \Rightarrow x = 2$.

Equations of Circles

• Standard Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$.

  • $(h, k)$ represent the coordinates of the center.

  • $r$ is the radius.

• Midpoint Formula: Used to find the center if diameter endpoints are known: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

• Distance Formula: Used to find the radius (distance between center and any point on the circle): $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

• Equation Examples:

  • Center $(0, 0)$, radius $2.5 \Rightarrow x^2 + y^2 = 6.25$.

  • Center $(-2, 5)$, radius $7 \Rightarrow (x + 2)^2 + (y - 5)^2 = 49$.

• Quadratic Formula Integration: In segment problems leading to quadratics ($ax^2 + bx + c = 0$), use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Practical Applications and Word Problems

• Northern Lights: Calculates the visible portion of Earth from a light flash at a specific altitude. Earth radius is approx $\text{4000 miles}$. Flash occurs $150\text{ miles}$ above point $C$. Calculations involve finding the central angle using trigonometry and the secant/tangent theorems.

• Mount Rainier Observation: Determining the visible arc of the Earth from an elevation of $2.73\text{ miles}$.

• Earthquake Epicenter: The epicenter is located by finding the intersection point of three circles, where each circle represents the distance from a seismograph station ($A$, $B$, and $C$).

  • Station $A$: Center $(-2, 2.5)$, radius $7$.

  • Station $B$: Center $(4, 6)$, radius $4$.

  • Station $C$: Center $(3, -2.5)$, radius $5$.

Questions & Discussion

• Identifying Segments: The class discussed the best descriptors for lines in $\bigodot C$. Segment $\overline{AG}$ was classified as a tangent ray while $\overline{DE}$ was labeled as a chord.

• Circumscribed Angle Logic: Student examined the relationship between circumscribed angles and central angles, concluding that $\angle ADB = 180^\circ - \angle ACB$.

• Tangent Circles Logic: Discussion focused on "common internal" vs "common external" tangents. A common internal tangent must cross the segment connecting the two centers.