### 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.

**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

- For example, when finding the coordinates of

### 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
*PQ*^{2}+*QR*^{2 }=*PR*^{2}. - Part C. Compare your right triangle with a partner.

Instructional Task 2 (MTR.3.1)

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.

- Part A. What are possible coordinates of P if

Instructional Task 3 (

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

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

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Problem-Solving Tasks

## 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.

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.

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

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.

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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task