Geometry R - Unit 13: Circles Study Guide

Course Schedule and Curricular Timeline

• Unit 13 Name: Circles.
• Constructions Project Timeline:   - Assigned: Wednesday, March 28.   - Classroom work days: March 28, March 29.   - Due Date: Tuesday, April 24.
• Instructional Progression:   - Monday 4/9: Central angles and Inscribed Angles (Day 1).   - Tuesday 4/10: Central angles and Inscribed Angles (Day 2).   - Wednesday 4/11: Arcs and Chords.   - Thursday 4/12: Area of Sectors, Area problems, and Arc Length.   - Friday 4/13: Tangents and Unit 14 Quiz 1.   - Monday 4/16: Secants, Tangents, and Angle Measures.   - Tuesday 4/17: Lengths of Secant Segments, Tangent Segments, and Chords.   - Wednesday 4/18: Writing Equations of Circles (Standard Form) and Unit 14 Quiz 2.   - Thursday 4/19: Equations of Circles (General Form).   - Friday 4/20: Inscribed and Circumscribed Polygons and Unit 14 Quiz 3.   - Monday 4/23 - Tuesday 4/24: Review Sheet and Review Sessions.   - Wednesday 4/25: Unit 14 Test.

Fundamentals of Circles and Angles

• Definitions:   - Circle: The locus or set of all points in a plane equidistant from a given point called the center of the circle.   - Central Angle: An angle whose vertex is the center of the circle. Its sides contain two radii of the circle.   - Congruent Circles: Two circles are congruent circles if and only if they have congruent radii.   - Concentric Circles: Coplanar circles that share the same center.
• Central Angle Sum:   - The sum of the measures of the central angles of a circle with no interior points in common is $360$ degrees.   - Example Calculation: Find the value of $x$ given variables in central angle diagrams.

Arc Properties and Measure

• Arc Definition: An arc is a portion of a circle defined by two endpoints. A central angle separates the circle into two arcs with measures related to the measure of the central angle.
• Arc Classifications:   - Minor Arc: The shortest arc connecting two endpoints on a circle.   - Major Arc: The longest arc connecting two endpoints on a circle.   - Semicircle: An arc with endpoints that lie on a diameter.
• Measuring Arcs:   - The measure of an arc of a circle is equal to the measure of its corresponding central angle.   - Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.   - Adjacent Arcs: Arcs that have exactly one point in common.
• Congruent Arcs:   - Arcs in the same or congruent circles that have the same measure.   - In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
• Practice Scenarios:   - Identify major arcs, minor arcs, and semicircles given a diameter in Circle $K$.   - Calculate measures such as m\overparen{GJ}, m\overparen{GH}, m\overparen{GLH}, and m\overparen{GLJ}.

Inscribed Angles and Theorems

• Definitions:   - Inscribed Angle: An angle possessing a vertex on a circle and sides that contain chords of the circle.   - Intercepted Arc: An arc that has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle.
• Inscribed Angle Theorem:   - If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.   - Formula: $\text{Angle} = \frac{1}{2} \times \text{Measure of Intercepted Arc}$.
• Related Theorems:   - If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.   - Thales’ Theorem: An inscribed angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle.
• Guided Examples and Questions:   - Find $m\angle P$ and m\overparen{PO}.   - Solve for $x$ if $m\angle S = 3x$ and $m\angle V = x + 16$, find $m\angle S$.   - Given radius $17\,miles$ and a diameter $AB$. If $BC = 30\,miles$, find $AC$.   - The Boris Scenario: Rectangle $ABCD$ is inscribed in circle $P$. Boris says diagonal $AC$ could pass through the center, but it does not have to. (Evaluated through Thales' Theorem showing that a rectangle inscribed in a circle must have diagonals as diameters).

Relationships Between Arcs and Chords

• Basic Chord Theorems:   - In the same circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.   - In a circle, central angles are congruent if and only if their corresponding chords are congruent.   - Arcs between parallel chords are congruent.
• Bisection Theorems:   - If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc.   - If a diameter (or radius) of a circle bisects a chord, then it must be perpendicular to the chord.   - If a line, segment, or ray divides an arc into two congruent arcs, then it bisects the arc.
• Equidistant Chords:   - In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
• Analytical Examples:   - If circle $J \cong \text{circle } K$ and \overparen{MN} \cong \overparen{PQ}, find the length $PQ$.   - Finding chord segments in Circle $S$ or Circle $A$ using ratios such as m\overparen{BC} : m\overparen{BD} : m\overparen{EC} = 1:2:4.

Arc Length and Radian Measures

• Arc Length: The distance between endpoints along an arc measured in linear units. It is a fraction of the circumference.
• Arc Length Proportion:   - $\frac{\text{Length of Arc } L}{\text{Circumference}} = \frac{\text{Degree Measure of Arc}}{360}$.
• Radian Measure:   - A radian is the measure of the central angle of a sector of a circle with arc length of one radius length.   - Relationship: $\pi \text{ radians} = 180^{\circ}$.   - Degree to Radian Conversion: Multiply degrees by $\frac{2\pi}{360}$ or $\frac{\pi}{180}$.   - Example: Convert $120^{\circ}$ to radians ($\frac{2\pi}{3} \text{ radians}$).

Area of Sectors

• Definitions:   - Circle Area: $A = \pi r^2$.   - Sector: A pie-shaped portion of the circle enclosed by 2 radii and the edge of the circle.
• Sector Area Formula:   - $\frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{\theta}{360}$, where $\theta$ is the central angle measure.
• Applications:   - Find area if $r = 10\,cm$ and angle is $72^{\circ}$.   - Find area if $r = 5\,in$ and angle is $60^{\circ}$.   - Work backward: If area is $15\pi\,cm^2$ and $r = 5\,cm$, find the central angle.   - Sector that is $\frac{1}{3}$ the area of the circle (central angle is $120^{\circ}$).
• Complex Geometry Example: Equilateral triangle $\Delta ABC$ with side $20\,cm$. Vertices are centers of circles creating arcs through midpoints. Find sector area, triangle area, shaded region area, and shaded perimeter.

Tangents in Circles

• Tangent Definition: A line in the same plane as a circle that intersects the circle in exactly one point, the point of tangency.
• Common Tangent: A line, ray, or segment tangent to two circles in the same plane.
• Perpendicularity Theorem: In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.
• Congruent Tangents Theorem: If two segments from the same exterior point are tangent to a circle, then they are congruent.
• Circumscribed Polygons: A polygon where every side is tangent to a circle.   - Example: Find $x$ for quadrilateral $RSTU$ circumscribed about circle $J$ with perimeter $18$.

Secants and Angle Measures

• Secant Definition: A line that intersects a circle in exactly two points.
• Interior Intersecting Secants: If two secants or chords intersect in the interior of a circle, the measure of the angle formed is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.   - Formula: $\text{Angle} = \frac{1}{2} \times (\text{arc}_1 + \text{arc}_2)$.
• Tangent-Secant Intersection at Point of Tangency: If a secant and a tangent intersect at the point of tangency, the measure of each angle formed is half the measure of its intercepted arc.
• Exterior Intersections: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, the angle formed is half the difference of the intercepted arcs.   - Formula: $\text{Angle} = \frac{1}{2} \times (\text{Major Arc} - \text{Minor Arc})$.

Power of Points and Segment Lengths

• Intersecting Chords Theorem: If two chords intersect in a circle, the products of the lengths of the chord segments are equal.   - Formula: $a \times b = c \times d$.
• Intersecting Secants Theorem (Exterior): If two secants intersect in the exterior of a circle, the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.   - Formula: $\text{whole} \times \text{external} = \text{whole} \times \text{external}$.
• Tangent-Secant Theorem (Exterior): If a tangent and a secant intersect in the exterior of a circle, the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.   - Formula: $\text{tangent}^2 = \text{whole} \times \text{external}$.

Equations of Circles

• Standard Form (Center-Radius Form): $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Derivation: Based on the Distance Formula, as all points on the circle are equidistant from the center.
• General Form and Conversion:   - General form equations like $x^2 + 4x + y^2 - 12y - 41 = 0$ must be rewritten in standard form by completing the square for both $x$ and $y$.
• Graphing and Identification:   - Identify a circle, a point, or an empty set based on the result of the conversion.   - Example: Solve for the center and radius of $x^2 - 20x + y^2 + 6y = 35$.

Inscribed Quadrilaterals

• Theorem: If a quadrilateral is inscribed in a circle (called a cyclic quadrilateral), then its opposite angles are supplementary ($\text{Sum} = 180^{\circ}$).
• Algebraic Verification: Prove $x + y = 180^{\circ}$ in a cyclic quadrilateral diagram.

Geometric Constructions

• Inscribe an Equilateral Triangle:   1. Draw diameter $AB$ in circle $O$.   2. Center at $A$, radius $AO$, draw an arc intersecting the circle at $C$ and $D$.   3. Connect $B$, $C$, and $D$.
• Inscribe a Regular Hexagon:   1. Draw diameter $AB$ in circle $O$.   2. Use $A$ and $B$ as centers, radius $AO$, draw arcs intersecting circle at $C$, $D$, $E$, and $F$.   3. Connect $A$, $E$, $F$, $B$, $C$, and $D$.
• Inscribe a Square:   1. Draw diameter $AB$ in circle $O$.   2. Construct diameter $CD$ as the perpendicular bisector of $AB$.   3. Connect $A$, $D$, $B$, and $C$.
• Inscribe a Regular Octagon:   1. Locate points $A$, $B$, $C$, and $D$ as per square construction.   2. Draw $BC$ and construct perpendicular bisector $EF$.   3. Use $A$ and $B$ as centers, radius $BE$, draw arcs intersecting circle at $G$, $H$, and $I$.   4. Connect $A$, $I$, $C$, $E$, $B$, $G$, $D$, and $H$.