# Cluster 1: Understand and apply theorems about circles. (Geometry - Additional Cluster)

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.G-C.1
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Geometry: Circles

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-C.1.AP.1a
Compare the ratio of diameter to circumference for several circles to establish all circles are similar.
MAFS.912.G-C.1.AP.2a
Identify and describe relationships among inscribed angles, radii and chords.
MAFS.912.G-C.1.AP.3a
Construct the inscribed and circumscribed circles of a triangle.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Assessments

Sample 1 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 3 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

## Educational Games

Circle Up!:

This interactive game helps you learn about angles and segments, lines and arcs in a circle and how they are related. You will compete against yourself and earn points as you answer questions about radius, diameter, chord, tangent line, central angles and inscribed angles and intercepted arcs.

Type: Educational Game

Tangled Web: An Angle Relationships Game:

You are a robotic spider tangled up in an angular web. Use your knowledge of angle relationships to collect flies and teleport through wormholes to rescue your spider family!

Type: Educational Game

## Formative Assessments

Central and Inscribed Angles:

Students are asked to describe the relationship between a central angle and an inscribed angle that intercept the same arc.

Type: Formative Assessment

Using a Compass To Construct a Tangent Line:

Students are asked to construct a tangent line to a given circle from a given exterior point.

Type: Formative Assessment

Students are asked to draw a circle, a tangent to the circle, and a radius to the point of tangency. Students are then asked to describe the relationship between the radius and the tangent line.

Type: Formative Assessment

Students are asked to prove that opposite angles of a quadrilateral, inscribed in a circle, are supplementary.

Type: Formative Assessment

Inscribed Angle on Diameter:

Students are asked to find the measures of two inscribed angles of a circle.

Type: Formative Assessment

Constructing a Tangent Line:

Students are asked to complete and justify the construction of a line tangent to a circle from an exterior point.

Type: Formative Assessment

Similar Circles:

Students are given two circles with different radii and are asked to prove that the circles are similar.

Type: Formative Assessment

Circles with Angles:

Students are given a diagram with inscribed, central, and circumscribed angles and are asked to identify each type of angle, determine angle measures, and describe relationships among them.

Type: Formative Assessment

All Circles Are Similar:

Students are given two circles with different radius lengths and are asked to prove that the circles are similar.

Type: Formative Assessment

Inscribed Circle Construction:

Students are asked to use a compass and straightedge to construct an inscribed circle of an acute scalene triangle.

Type: Formative Assessment

Circumscribed Circle Construction:

Students are asked to use a compass and straightedge to construct a circumscribed circle of an acute scalene triangle.

Type: Formative Assessment

## Lesson Plans

Why are Circles Similar?:

Type: Lesson Plan

The Seven Circles Water Fountain :

This lesson provides an opportunity for students to apply concepts related to circles, angles, area, and circumference to a design situation.Â

Type: Lesson Plan

Are All Circles Similar? :

This lesson allows students to prove that all circles are similar using transformations. Students will need prior knowledge of similarity, transformations, and the definition of a circle. The lesson begins with a warm up regarding dilations, then poses the question: Are all circles similar? The students are guided through the proof using a translation and dilation. The teacher emphasizes the details in the proof. The lesson closes with an exit ticket.

Type: Lesson Plan

Circle to Circle:

Students use coordinate based translations and dilations to prove circles are similar.

Type: Lesson Plan

Geometry Problems: Circles and Triangles:

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties:

• Solving problems by determining the lengths of the sides in right triangles.
• Finding the measurements of shapes by decomposing complex shapes into simpler ones.

The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.

Type: Lesson Plan

Geometry Problems: Circles and Triangles:

This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties solving problems by determining the lengths of the sides in right triangles and finding the measurements of shapes by decomposing complex shapes into simpler ones. The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.

Type: Lesson Plan

Off on a Tangent:

Students learn and apply vocabulary, notation, concepts, and geometric construction techniques associated with circles and their tangents to a historical real-world scenario, the Mason-Dixon Line, and a hypothetical real-world scenario, the North-South Florida Line.

Type: Lesson Plan

Inscribing and Circumscribing Right Triangles:

This lesson is designed to enable students to develop strategies for describing relationships between right triangles and the radii of their inscribed and circumscribed circles.

Type: Lesson Plan

Seeking Circle Angles:

Students will start this lesson with a win-lose-draw game to review circle vocabulary words. They will then use examples on a discovery sheet to discover the relationships between arcs and the angles whose vertex is located on a circle, in the interior of the circle, and exterior to the circle. They will wrap up the lesson in a class discussion and questions answered on white boards.

Type: Lesson Plan

Seeking Circle Segments:

Students will start this lesson with a "Pictionary" game to review circle vocabulary terms. They will then use computers and GeoGebra to discover the relationships between portions of segments that intersect in the interior of the circle, and exterior to the circle. They will wrap up the lesson in a class discussion and consensus on rules (formulas).

Type: Lesson Plan

## Original Student Tutorials

Pennant Company Challenge: Inscribed Circles of Triangles:

Discover how easy it is for Katie to construct an inscribed circular logo on her company's triangular pennant template.  If she completes the task first, she will win a \$1000 bonus! Follow along with this interactive tutorial.

Type: Original Student Tutorial

Good as New:

Learn the steps to circumscribe a circle around a triangle in this interactive tutorial about constructions. Grab a compass, straightedge, pencil and paper to follow along!

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

All Circles Are Similar- Especially Circular Pizza!:

What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations.

Type: Perspectives Video: Professional/Enthusiast

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Tangent to a circle from a point:

This problem solving task challenges students to describe and compare different angles.

Right triangles inscribed in circles II:

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Locating Warehouse:

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

## Tutorial

Constructing Tangent Lines:

This GeoGebraTube interactive worksheet shows you the step by step method for constructing tangent lines to a circle from a point outside the circle.  Use the slider to see each step, and read below the illustration to follow the steps.

Type: Tutorial

## Virtual Manipulative

Circumscribe a Circle About a Triangle:

In this GeoGebraTube interactive worksheet, you can watch the step by step process of circumscribing a circle about a triangle.  Using paper and pencil along with this resource will reinforce the concept.

Type: Virtual Manipulative

## Worksheet

Inscribing a circle in a triangle I:

This problem solving task shows how to inscribe a circle in a triangle using angle bisectors.

Type: Worksheet

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

Pennant Company Challenge: Inscribed Circles of Triangles:

Discover how easy it is for Katie to construct an inscribed circular logo on her company's triangular pennant template.  If she completes the task first, she will win a \$1000 bonus! Follow along with this interactive tutorial.

Type: Original Student Tutorial

Good as New:

Learn the steps to circumscribe a circle around a triangle in this interactive tutorial about constructions. Grab a compass, straightedge, pencil and paper to follow along!

Type: Original Student Tutorial

## Educational Game

Circle Up!:

This interactive game helps you learn about angles and segments, lines and arcs in a circle and how they are related. You will compete against yourself and earn points as you answer questions about radius, diameter, chord, tangent line, central angles and inscribed angles and intercepted arcs.

Type: Educational Game

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Tangent to a circle from a point:

This problem solving task challenges students to describe and compare different angles.

Right triangles inscribed in circles II:

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Locating Warehouse:

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.

## Tutorial

Constructing Tangent Lines:

This GeoGebraTube interactive worksheet shows you the step by step method for constructing tangent lines to a circle from a point outside the circle.  Use the slider to see each step, and read below the illustration to follow the steps.

Type: Tutorial

## Virtual Manipulative

Circumscribe a Circle About a Triangle:

In this GeoGebraTube interactive worksheet, you can watch the step by step process of circumscribing a circle about a triangle.  Using paper and pencil along with this resource will reinforce the concept.

Type: Virtual Manipulative

## Worksheet

Inscribing a circle in a triangle I:

This problem solving task shows how to inscribe a circle in a triangle using angle bisectors.

Type: Worksheet

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Tangent to a circle from a point:

This problem solving task challenges students to describe and compare different angles.

Right triangles inscribed in circles II:

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Placing a Fire Hydrant:

This problem solving task asks students to place a fire hydrant so that it is equal distance from three given points.

Locating Warehouse:

This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).

Inscribing a triangle in a circle:

This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle.

Circumcenter of a triangle:

This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Inscribing a circle in a triangle II:

This problem solving task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.