# Standard 5: Make formal geometric constructions with a variety of tools and methods.

General Information
Number: MA.912.GR.5
Title: Make formal geometric constructions with a variety of tools and methods.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.912.GR.5.AP.1
Construct a copy of a segment.
MA.912.GR.5.AP.2
Construct the bisector of a segment, including the perpendicular bisector of a line segment.
MA.912.GR.5.AP.3
Select the inscribed and circumscribed circles of a triangle.

## Related Resources

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

## Formative Assessments

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

Constructing a Congruent Angle:

Students are asked to construct an angle congruent to a given angle.

Type: Formative Assessment

The Sprintersâ€™ Race:

Students are given a grid with three points (vertices of a right triangle) representing the starting locations of three sprinters in a race and are asked to determine the center of the finish circle, which is equidistant from each sprinter.

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

Constructing a Congruent Segment:

Students are asked to construct a line segment congruent to a given line segment.

Type: Formative Assessment

Regular Hexagon in a Circle:

Students are asked to construct a regular hexagon inscribed in a circle.

Type: Formative Assessment

Equilateral Triangle in a Circle:

Students are asked to construct an equilateral triangle inscribed in a circle.

Type: Formative Assessment

Construct the Center of a Circle:

Students are asked to construct the center of a circle.

Type: Formative Assessment

Square in a Circle:

Students are asked to construct a square inscribed in a circle.

Type: Formative Assessment

Bisecting a Segment and an Angle :

Students are asked to construct the bisectors of a given segment and a given angle and to justify one of the steps in each construction.

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

Constructions for Perpendicular Lines:

Students are asked to construct a line perpendicular to given line (1) through a point not on the line and (2) through a point on the line.

Type: Formative Assessment

## Lesson Plans

Geometric Construction Site:

This lesson takes students from simple construction of line segments and angles to an optional extension worksheet for creating triangles.

Type: Lesson Plan

Keeping Triangles in Balance: Discovering Triangle Centroid is Concurrent Medians:

In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the centroid is a point of concurrency for the medians of a triangle and recognize its associated usage with the center of gravity or barycenter. This set of instructional materials provides the teacher with hands-on activities using technology as well as paper-and-pencil methods.

Type: Lesson Plan

Construction of Inscribed Regular Hexagon:

A GeoGebra lesson for students to become familiar with computer based construction tools. Students work together to construct a regular hexagon inscribed in a circle using rotations. Directions for both a beginner and advanced approach are provided.

Type: Lesson Plan

Copying and Bisecting an Angle:

This lesson is a gradual release model for constructing congruent angles and bisecting angles.

Type: Lesson Plan

Inscribe Those Rims:

This lesson will engage students with an interactive and interesting way to learn how to inscribe polygons in circles.

Type: Lesson Plan

Bisecting Angles And Line Segments:

This construction lesson will teach students how to bisect an angle and how to find the perpendicular bisector of a segment using a compass and straightedge.

Type: Lesson Plan

Construction Junction:

Students will learn how to construct an equilateral triangle and a regular hexagon inscribed in a circle using a compass and a straightedge.

Type: Lesson Plan

Triangle Medians:

This lesson will have students exploring different types of triangles and their medians. Students will construct mid-points and medians to determine that the medians meet at a point.

Type: Lesson Plan

Inscribe it:

This activity allows students to practice the construction process inscribing a regular hexagon and an equilateral triangle in a circle using GeoGebra software.

Type: Lesson Plan

Construct Regular Polygons Inside Circles:

Students will be able to demonstrate that they can construct, using the central angle method, an equilateral triangle, a square, and a regular hexagon, inscribed inside a circle, using a compass, straightedge, and protractor. They will use worksheets to master the construction of each polygon, one inside each of three different circles. As an extension to this lesson, if computers with GeoGebra are available, the students should be able to perform these constructions on computers as well.

Type: Lesson Plan

Construct This:

In this lesson, students will construct a square inscribed in a circle using the properties of a square and determine if there is more than one way to complete the construction.

Type: Lesson Plan

I Am Still Me Transformed.:

Students explore ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, for them to develop a better understanding of symmetries in transformational geometry.

Type: Lesson Plan

Constructing an Angle Bisector:

Students construct an angle bisector given a straightedge and compass then verify their process. The Guided Practice is done in stations. One that is teacher-led and one that is student-led. In order to complete the student-led Guided Practice, access to a teacher computer and projector is needed. Then the students independently create their own angle and its bisector and verify their work for a grade. Students use patty paper and protractors to confirm the accuracy of the construction.

Type: Lesson Plan

Determination of the Optimal Point:

Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.

Type: Lesson Plan

Back to the Basics: Constructions:

Students will use a compass and straightedge to develop methods for constructions. GeoGebra directions are also provided.

Type: Lesson Plan

Sage and Scribe - Points, Lines, and Planes:

Students will practice using precise definitions while they draw images of Points, Lines, and Planes. Students will work in pairs taking turns describing an image while their partner attempts to accurately draw the image.

Type: Lesson Plan

Fundamental Property of Reflections:

This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.

Type: Lesson Plan

Paper Plate Origami:

A hands-on activity where students construct inscribed regular polygons in a circle using models. Through guided questions, students will discover how to divide a model (paper plate) into 3, 4, and 6 parts. Using folding, a straightedge, and a compass, they will construct an equilateral triangle, a square, and a regular hexagon in their circles.

Type: Lesson Plan

What's the Point? Part 2:

In this lesson, students use a paper-folding technique to discover the properties of angle bisectors. At the conclusion of the activity, students will be able to compare/contrast the points of concurrency of perpendicular and angle bisectors.

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

Circumnavigating the Circumcenter:

Students use the concurrent point of perpendicular bisectors of triangle sides to determine the circumcenter of three points. Students will reason that the circumcenter of the vertices of a polygon is the optimal location for placement of a facility to service all of the needs of sites at the vertices forming the polygon.

Type: Lesson Plan

Crafty Circumference Challenge:

Students learn about geometric construction tools and how to use them. Students will partition the circumference of a circle into three, four, and six congruent arcs which determine the vertices of regular polygons inscribed in the circle. An optional project is included where students identify, find, and use recycled, repurposed, or reclaimed objects to create "crafty" construction tools.

Type: Lesson Plan

Right turn, Clyde!:

Students will develop their knowledge of perpendicular bisectors & point of concurrency of a triangle, as well as construct perpendicular bisectors through real world problem solving with a map.

Type: Lesson Plan

Halfway to the Middle!:

Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments.

Type: Lesson Plan

Location, Location, Location, Location?:

Students will use their knowledge of graphing concurrent segments in triangles to locate and identify which points of concurrency are associated by location with cities and counties within the Texas Triangle Mega-region.

Type: Lesson Plan

St. Pi Day construction with a compass & ruler:

St. Pi Day construction with compass

This activity uses a compass and straight-edge(ruler) to construct a design. The design is then used to complete a worksheet involving perimeter, circumference, area and dimensional changes which affect the scale factor ratio.

Type: Lesson Plan

Concurrent Points Are Optimal:

Students will begin with a review of methods of construction of perpendicular bisectors and angle bisectors for the sides of triangles. Included in the review will be a careful discussion of the proofs that the constructions actually produce the lines that were intended.

Next, students will investigate why the perpendicular bisectors and angle bisector are concurrent, that is, all three meet at a single meet.

A more modern point of currency is the Fermat-Torricelli point (F-T). The students will construct (F-T) in GeoGebra and investigate limitations of its existence for various types of triangles.

Then a set of scenarios will be provided, including some one-dimensional and two-dimensional situations. Students will use GeoGebra to develop conjectures regarding whether a point of concurrency provides the solution for the indicated situation, and which one.

A physical model for the F-T will be indicated. The teacher may demonstrate this model but that requires three strings, three weights, and a base that has holes. A recommended base is a piece of pegboard (perhaps 2 feet by 3 feet), the weights could be fishing weights of about 3 oz., the string could be fishing line; placing flexible pieces of drinking straws in the holes will improve the performance.

The combination of geometry theorems, dynamic geometry software, a variety of contexts, and a physical analog can provide a rich experience for students.

Type: Lesson Plan

What's the Point? Part 1:

This is a patty paper-folding activity where students measure and discover the properties of the point of concurrency of the perpendicular bisectors of the sides of a triangle.

Type: Lesson Plan

## Original Student Tutorials

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

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

Meet Me Half Way:

Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.

Type: Original Student Tutorial

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in 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

The Blueprints of Construction:

Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Teaching Ideas

Geometric Constructions- Angle Bisector:

Unlock an effective teaching strategy for making connections in geometric constructions in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Perpendicular Bisectors & Constructions:

Unlock an effective teaching strategy for teaching geometric constructions, specifically perpendicular bisectors, in this Teacher Perspectives video for educators.

Type: Perspectives Video: Teaching Idea

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Construction of perpendicular bisector:

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent to a circle from a point:

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

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.

Bisecting an angle:

This problem solving task challenges students to bisect a given angle.

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.

Angle bisection and midpoints of line segments:

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

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.

## Student Resources

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

## Original Student Tutorials

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

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

Meet Me Half Way:

Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.

Type: Original Student Tutorial

A Square Peg in a Round Hole:

Learn how to construct an inscribed square in a circle and why certain constructions are used in this interactive tutorial.

Type: Original Student Tutorial

Designing with Hexagons:

Learn how to construct an inscribed regular hexagon and equilateral triangle in a circle in 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

The Blueprints of Construction:

Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.

Type: Original Student Tutorial

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Construction of perpendicular bisector:

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent to a circle from a point:

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

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.

Bisecting an angle:

This problem solving task challenges students to bisect a given angle.

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.

Angle bisection and midpoints of line segments:

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

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.

## Parent Resources

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

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Inscribing a hexagon in a circle:

This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

Construction of perpendicular bisector:

This problem solving task challenges students to construct a perpendicular bisector of a given segment.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent to a circle from a point:

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

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.

Bisecting an angle:

This problem solving task challenges students to bisect a given angle.

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.

Angle bisection and midpoints of line segments:

This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.

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.