# MA.912.GR.3.2

Given a mathematical context, use coordinate geometry to classify or justify definitions, properties and theorems involving circles, triangles or quadrilaterals.

### Examples

Example: Given Triangle ABC has vertices located at (-2,2), (3,3) and (1,-3), respectively, classify the type of triangle ABC is.

Example: If a square has a diagonal with vertices (-1,1) and (-4,-3), find the coordinate values of the vertices of the other diagonal and show that the two diagonals are perpendicular.

### Clarifications

Clarification 1: Instruction includes using the distance or midpoint formulas and knowledge of slope to classify or justify definitions, properties and theorems.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Circle
• MA.912.GR.7.2
• Slope
• Triangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8 and Algebra 1, students used coordinate systems to study lines and the find distances between points. In Geometry, students expand their knowledge of coordinate geometry to further study lines and distances and relate them to classifying geometric figure. In later courses, coordinates will be used to study a variety of figures, including conic sections and shapes that can be studied using polar coordinates.
• Instruction includes the connection of the Pythagorean Theorem (as was used in grade 8) to the distance formula. It is important that students not depend on just the memorization of the distance formula.
• Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form) or with approximations (e.g., rounding to the nearest tenth or hundredth). It is also important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts.
• In this benchmark, instruction is related to circles, triangles and quadrilaterals, their definitions and properties. It may be helpful to review these definitions and properties and the different types of triangles and quadrilaterals as it was part of instruction within elementary grades.
• Instruction includes determining when slope criteria may be necessary.
• For example, when classifying triangles and quadrilaterals or finding side lengths, the slope criteria may be needed.
• For example, determining sides of equal measures will decide if a triangle is isosceles or if a quadrilateral is a rhombus.
• For example, the slope criteria for parallel lines may help when deciding if a quadrilateral is a parallelogram.
• For example, the slope criteria for perpendicular lines may help when deciding if a triangle is right or if a quadrilateral is a rectangle.
• Explore with the students different approaches for the same goal.
• For example, given a parallelogram they can determine if it is a rectangle using the slope criteria to identify right angles or using the distance formula (or the Pythagorean Theorem) to identify if the diagonals are congruent.
• Instruction includes opportunities for students to find the coordinates of missing vertices of a triangle or quadrilateral using coordinate geometry and applying definitions, properties, or theorems.
• For example, when finding the coordinates of P such that PQRS is a rhombus (given the coordinates of Q, R and S), guide the students to plot the points on the coordinate plane and make a conjecture about the location of P. Have students determine if their conjectures are true. Additionally, have students discuss the definitions or properties they may use in each case

### Common Misconceptions or Errors

• Students may use imprecise methods or incomplete definitions to classify figures.

• Part A. What are the coordinates of P if PQRS is a right triangle and Q(−1, 2) and R(3, 0)?
• Part B. Show that PQ2+ QR= PR2
• Part C. Compare your right triangle with a partner.

• Three vertices of quadrilateral PQRS are at the points Q(−2, 1), R(3,−1) and S(−2,−3).
• Part A. What are possible coordinates of P if PQRS is a parallelogram?
• Part B. Show that PR bisects QS.
• Part C. Justify that PQRS is a parallelogram.

• Coordinates for three two-dimensional figures are given.
Figure A (2,3), (3,−4), (3,−2)
Figure B (3,3), (2, −1), (−2,0), (−1,4)
Figure C (−2,3), (−3,1), (0,−4), (3,2)
• Part A. Plot the points on the coordinate plane.
• Part B. Write a conjecture about the specific name of each two-dimensional figure. What would you need to determine your conjectures are true?
• Part C. Classify each figure.

### Instructional Items

Instructional Item 1
• Points A (0,2) and B (2,0) are endpoints of segment AB, the side of quadrilateral ABCD. List possible coordinates for points C and D if quadrilateral ABCD is a rhombus, not a square.

Instructional Item 2
• Given quadrilateral ABCD with vertices (−3,−4), (1,5), (5,3), and (5, −8), respectively, classify the type of quadrilateral.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.3.AP.2: Use coordinate geometry to classify definitions, properties and theorems involving circles, triangles, or quadrilaterals.

## Related Resources

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

## Formative Assessments

Midpoints of Sides of a Quadrilateral:

Students are asked to prove that the quadrilateral formed by connecting the midpoints of the sides of a given quadrilateral is a parallelogram.

Type: Formative Assessment

Type of Triangle:

Students are given the coordinates of three vertices of a triangle and are asked to use algebra to determine whetherÂ the triangle is scalene, isosceles, or equilateral.

Type: Formative Assessment

Diagonals of a Rectangle:

Students are given the coordinates of three of the four vertices of a rectangle and are asked to determine the coordinates of the fourth vertex and show the diagonals of the rectangle are congruent.

Type: Formative Assessment

Students are given the coordinates of the vertices of a quadrilateral and are asked to determine whether the quadrilateral could also be a parallelogram, rhombus, rectangle, square, or trapezoid.

Type: Formative Assessment

Triangle Midsegment Proof:

Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Type: Formative Assessment

## Lesson Plans

Basic Definitions in Geometry:

A set of basic definitions in geometry (line segment, ray, angle, perpendicular lines, and parallel lines) is addressed. The notation used in naming each defined term is also emphasized.

Type: Lesson Plan

Triangle Mid-Segment Theorem:

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

Proof of Quadrilaterals in Coordinate Plane:

This lesson is designed to instruct students on how to identify special quadrilaterals in the coordinate plane using their knowledge of distance formula and the definitions and properties of parallelograms, rectangles, rhombuses, and squares. Task cards, with and without solution-encoded QR codes, are provided for cooperative group practice. The students will need to download a free "QR Code Reader" app onto their SmartPhones if you choose to use the cards with QR codes.

Type: Lesson Plan

To Be or Not to Be a Parallelogram:

Students apply parallelogram properties and theorems to solve real world problems. The acronym, P.I.E.S. is introduced to support a problem solving strategy, which involves drawing a Picture, highlighting important Information, Estimating and/or writing equation, and Solving problem.

Type: Lesson Plan

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

Type: Lesson Plan

Musical Chairs with Words and a Ball:

This lesson introduces students to concepts and skills that they will use throughout the year. Students will learn that the terms point, and line are considered "undefined." Students will play musical chairs while learning to develop precise definitions of circle, angle, parallel line, and perpendicular line, using counterexamples at different classroom stations. Students will identify models, use notation, and make sketches of these terms.

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

Pondering Points Proves Puzzling Polygons:

In a 55 minute class, students use whiteboards, Think-Pair-Share questioning, listen to a quadrilateral song, and work individually and in groups to learn about and gain fluency in using the distance and slope formulas to prove specific polygon types.

Type: Lesson Plan

Proving Parallelograms Algebraically:

This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram.

Type: Lesson Plan

Proving quadrilaterals algebrically using slope and distance formula:

Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then justify their proofs to their classmates.

Type: Lesson Plan

In this lesson, students will use coordinates to algebraically prove that quadrilaterals are rectangles, parallelograms, and trapezoids. A through introduction to writing coordinate proofs is provided as well as plenty of practice.

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

This lesson provides a series of assignments for students at the Getting Started, Moving Forward, and Almost There levels of understanding for the Mathematics Formative Assessment System (MFAS) Task Describe the Quadrilateral (CPALMS Resource ID#59180). The assignments are designed to "move" students from a lower level of understanding toward a complete understanding of writing a coordinate proof involving quadrilaterals.

Type: Lesson Plan

Ellipse Elements and Equations:

Students will write the equation of an ellipse given foci and directrices using graphic and analytic methods.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

Polygon...Prove it:

While this is an introductory lesson on the standard, students will enjoy it, as they play "Speed Geo-Dating" during the Independent practice portion. Students will use algebra and coordinates to prove rectangles, rhombus, and squares. Properties of diagonals are not used in this lesson.

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

Let's Prove the Pythagorean Theorem:

Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.

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

Going the Distance:

This lesson uses the Pythagorean Theorem to derive several iterations of the Distance Formula. The Distance Formula is then used to calculate the distance between two points on both directional maps and the Cartesian coordinate plane. Vocabulary relating to vectors is also introduced.

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

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

## MFAS Formative Assessments

Students are given the coordinates of the vertices of a quadrilateral and are asked to determine whether the quadrilateral could also be a parallelogram, rhombus, rectangle, square, or trapezoid.

Diagonals of a Rectangle:

Students are given the coordinates of three of the four vertices of a rectangle and are asked to determine the coordinates of the fourth vertex and show the diagonals of the rectangle are congruent.

Midpoints of Sides of a Quadrilateral:

Students are asked to prove that the quadrilateral formed by connecting the midpoints of the sides of a given quadrilateral is a parallelogram.

Triangle Midsegment Proof:

Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Type of Triangle:

Students are given the coordinates of three vertices of a triangle and are asked to use algebra to determine whetherÂ the triangle is scalene, isosceles, or equilateral.

## Student Resources

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

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

## Parent Resources

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

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Mt. Whitney to Death Valley:

This task engages students in an open-ended modeling task that uses similarity of right triangles.