**Math Topics**

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Select the cluster for resources on this page:

**Standards**:

- C-A.1. Prove that all circles are similar.
- C-A.2. Identify and describe relationships
among inscribed angles, radii, and chords.
*Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.* - C-A.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- C-A.4. (+) Construct a tangent line from a point outside a given circle to the circle.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with C-A and C-B:

CK-12:Geometry: concepts, videos, and practice exercises for learners to complete: Use with C-A.2.

- Inscribed Angles in Circles
- Tangent Lines: Lines perpendicular to the radius drawn to the point of tangency.
- Chords: Arcs determined by angles whose vertex is the center of a circle and chords (segments that connect two points on a circle).
- Angles On and Inside a Circle
- Angles Outside a Circle

Khan Academy: Geometry: Circles:

- Proof: all circles are similar: video. Aligns with C-A.1.

Khan Academy: Geometric Constructions, Tutorial videos:

- Construct a circle inscribed and circumscribed about a triangle: Aligns with C-A.3.
- Constructing a line tangent to a circle: 2 videos: given P on the circle, and P outside of the circle. Aligns with C-A.4.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

- Lesson set: Prove that all circles are similar: Four video lessons: Similarity is demonstrated in these using similar triangles, translations and dilations. Aligns with C-A.1.
- Lesson set: Identify and describe relationships among inscribed angles, radii, and chords: Four video lessons: Find measures of inscribed angles using central angles, understand tangents are perpendicular to radii, find relationships between central angles and circumscribed angles, find missing angle measurements within a circle using the intersecting chord theorem. Aligns with C-A.2.
- Lesson set: Construct the inscribed and circumscribed circles of a triangle; prove properties of angles for a quadrilateral inscribed in a circle: Four video lessons: Construct circumscribed, inscribed circle of a triangle; find circumcenters and incenters of a triangle; show angle relationships of cyclic quadrilaterals by using Central Angle Theorem. Aligns with C-A.3.

MathsIsFun.com: Constructions: This is a series of 17 animated geometric constructions, which learners would do with a compass, pencil, and straightedge only. Among those are circle constructions to find a center, tangent lines from an external point, inscribed and circumscribed circles for triangles, circle touching three points, and a pentagon construction. Aligns with C-A.3 and C-A.4.

Math Open Reference: Constructions: The site provides step-by-step animations of constructions with compass and straightedge. The list is comprehensive for lines, line segments, angles, triangles, circles, arcs, ellipses, polygons, and non-Euclidean constructions. The following align with C-A.3 and C-A.4:

- Incircle of a triangle
- Circumcircle of a triangle
- Tangent to a circle at a point
- Tangents through an external point

NCTM Illuminations:

- Incenter-Incircle: This applet allows you to find the incenter and incircle of a triangle, and provides a visual explanation of why it occurs where it does.
- Circumcircle: Applet to construct the circumcircle about a triangle.
- Simson Line: Applet to construct and explore the Simson line.

Thinking Mathematics: Tangent Theorems for Circles and the Pythagorean Theorem: A YouTube video on two results from the geometry of circles that are easily proven using the Pythagorean theorem.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 21 manipulatives addressing C-A.2, 10 manipulatives for C-A.3, and 1 manipulative for C-A.4. Among those:

- Geometric elements of a circle: Aligns with C-A.2.
- Inscribed and central angles of a circle: Aligns with C-A.2.
- Inscribed angles that intercept the same arc: Aligns with C-A.2.
- Inscribed angles in a circle subtended by the same chord: Aligns with C-A.2.
- Thales' theorem: angles inscribed in a semi-circle: Aligns with C-A.2.
- Thales' theorem: a vector-based proof: Aligns with C-A.2.
- Equally distance chords: Aligns with C-A.2.
- Tangent chord angle: Aligns with C-A.2.
- Intersecting secants theorem: Aligns with C-A.2.
- Intersecting chords theorem: Aligns with C-A.2.
- The Chordal theorem: Aligns with C-A.2.
- The perpendicular bisector of a chord: Aligns with C-A.2.
- Sum of Opposite angles of a quadrilateral inscribed in a circle is 180 degrees: Aligns with C-A.2.
- Euclid, Book 3, Proposition 22: in a cyclic quadrilateral, opposite angles sum to 180 degrees: Aligns with C-A.2, C-A.3, and CO-C.10.
- Circles and triangles: Aligns with C-A.2.
- Three points determine a circle: Aligns with C-A.2.
- Reflection in a circle: Aligns with C-A.2.
- Tangent circles: Aligns with C-A.2.
- The bomb problem: This problem considers multiple points on a plane. In the "bomb problem", the smallest circle that contains all the points is found. The bomb problem comes from military tactics. What is the least powerful bomb that can destroy all targets, and where should it be dropped? Aligns with C-A.2, C-A.3.
- Triangle Calculator: The program calculates triangle centers: incenter I, centroid G, circumcenter O, orthocenter H. The points G, O, and H are collinear: Aligns with C-A.3.
- Problems on Circles IX: Circumcircles and Incircles of a Triangle: Aligns with C-A.3.
- Circumcircle and incircle of a triangle: Aligns with C-A.3.
- Johnson's theorem: Let three circles of equal diameter intersect at a point H and intersect pairwise at points A, B, and C. Then the circumcircle of the triangle ABC has the same diameter as the other circles. Aligns with C-A.3.
- Pairwise Tangent Circles Centered at the Vertices of a Triangle: Aligns with C-A.3.
- Tangents to a circle: Aligns with C-A.2, C-A.4.

**Multiple Choice:**

MathsIsFun.com: Circle Theorems: Content reviews inscribed angle theorems, angles in quadrilaterals inscribed in circles, the 90 degree angle between a tangent line and the radius drawn to the point of tangency. Examples are shown. Learners then complete 10 multiple choice problems on those theorems. Aligns with C-A.2 and C-A.3.

**Constructed-response:**

Khan Academy: HSG-G-C includes HSG-C.A.1 through HSG-C.A.4 Common Core aligned problems.

Also see Geometry: Circles: Practice questions with videos:

- Quiz: Inscribed Quadrilaterals: Aligns with C-A.2, C-A.3.
- Tangents of Circles Problems: involves central, inscribed, and circumscribed angles. Aligns with C-A.4.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Similar circles: Aligns with C-A.1.
- Right triangles inscribed in circles 1: Aligns with C-A.2.
- Inscribing a circle in a triangle 1: Aligns with C-A.3.

Inside Mathematics: MARS Tasks:

- Circle and Squares: Students calculate sides of squares given the radius of the inscribed and circumscribed circle and calculate ratios of nested circles and squares. This task aligns with standards C-A.2 and C-A.3.
- Circles in Triangles: Students use geometric properties of circles and triangles to prove that two triangles are congruent. This task aligns with geometry congruence standards CO-C.9 and CO-C.10 and geometry circle standards C-A.2 and C-A.3.

Mathematics Assessment Project: High School: Geometry:

- Task 403: Inscribing and Circumscribing Right Triangles
- Task 222: Geometry Problems: Circles and Triangles

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 7: Circles a Geometric Perspective: This module contains 10 classroom tasks. Standards C-A.1, C-A.2, C-A.3, and C-A.4 can be found within tasks 1-3 and task 6.

**Standards**:

- C-B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

**Technology-enhanced investigations:**

CK-12: Geometry: Arcs in Circles: concepts, videos, and practice problems for learners to complete.

Thinking Mathematics: Radian Measure: A YouTube video with explanation of the radian measure.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 10 manipulatives for C-B.5. Among those:

**Multiple Choice:**

MathsIsFun.com: Circle Sector and Segment: Content reviews arc length and area of a sector showing how formulas are derived. It also includes how to find the area of a segment. Learners can complete 10 multiple choice problems. Use with C-B.

**Constructed-response:**

Khan Academy: HSG-G-C includes HSG-C.B.5 Common Core aligned problems.

Also see Geometry: Circles Practice questions with videos:

- Arc measures in degrees: Aligns with C-B.5.
- Arc Length: Relate the length of the arc to the circumference and the central angle subtended by the arc in degrees. Aligns with C-B.5.
- Radians and Degrees: Convert angle measures from degrees to radians and vice versa. Aligns with C-B.5.
- Radians and Arc Length: Solve problems related to radians and arc length, like finding an arc length given a central angle in radians and a radius.
- Area of a sector: Relate the area of a sector to the area of the whole circle. Aligns with C-B.5.

**Performance tasks:**

Illustrative Mathematics: Geometry: Setting up sprinklers: This task aligns with standards in C-B.

Mathematics Assessment Project: High School: Geometry: Task 441: Sectors of Circles

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 7: Circles a Geometric Perspective: This module contains 10 classroom tasks. Standard C-B.5 can be found within tasks 7-9.

Common Core Math:
Intro | HS Geometry Domain: CO | SRT | **C** | GPE | GMD | MG |