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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.)
Grade: 912
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Circle
- Quadrilateral
- 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.
Instructional Tasks
Instructional Task 1 (MTR.2.1, MTR.4.1)- 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+ QR2 = PR2.
- Part C. Compare your right triangle with a partner.
Instructional Task 2 (MTR.3.1)
- 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.
Instructional Task 3 (MTR.3.1, MTR.4.1)
- 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.
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
MFAS Formative Assessments
Describe the Quadrilateral:
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 whether the triangle is scalene, isosceles, or equilateral.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
