# MA.912.GR.3.3

Use coordinate geometry to solve mathematical and real-world geometric problems involving lines, circles, triangles and quadrilaterals.

### Examples

Example: The line x+2y=10 is tangent to a circle whose center is located at (2,-1). Find the tangent point and a second tangent point of a line with the same slope as the given line.

Example: Given M(-4,7) and N(12,-1),find the coordinates of point P on so that P partitions in the ratio 2:3.

### Clarifications

Clarification 1: Problems involving lines include the coordinates of a point on a line segment including the midpoint.

Clarification 2: Problems involving circles include determining points on a given circle and finding tangent lines.

Clarification 3: Problems involving triangles include median and centroid.

Clarification 4: Problems involving quadrilaterals include using parallel and perpendicular slope criteria.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Circle
• Diameter
• 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 on their knowledge of coordinate geometry to solve problems geometric problems in real-world and mathematical contexts. In later courses, coordinates will be used to solve a variety of problems involving many shapes, including conic sections and shapes that can be studied using polar coordinates.
• Problem types include finding the midpoint of a segment (midpoint formula); partitioning a segment given endpoints and a ratio; writing the equation of a line, including lines that are parallel or perpendicular; finding the coordinates of the centroid of a triangle; and finding the distance between two points. In some cases, students may need to utilize systems of equations in order to determine solutions.
• Instruction includes the definition of a tangent to a circle and its properties, and the definition of the medians of a triangle and their point of concurrency (centroid).
• Instruction includes various approaches when finding the coordinates of a point partitioning a directed line segment (given the endpoints). (MTR.2.1MTR.3.1) Different methods are described below.
• The first concept that can be discussed is the connection to weighted average of two points. If the given ratio is a: b, that means the weights of the endpoints ($x$1 + $x$1) and ($x$2 + $y$2) are $\frac{\text{b}}{\text{a+b}}$ and $\frac{\text{a}}{\text{a+b}}$, respectively.
• For example, if the given ratio is 2: 3 and the points are at A(−3, 6) and B(4,−8) discuss with students where P is on its way from A to B. Students should be able to come with $\frac{\text{2}}{\text{5}}$. That means the weight of A is $\frac{\text{3}}{\text{5}}$ and the weight of B is $\frac{\text{2}}{\text{5}}$ . The next step is to calculate the $x$-coordinate of P using the weighted averages:  $x$p $\frac{\text{3}}{\text{5}}$x1 + $\frac{\text{2}}{\text{5}}$$x$2 which is equivalent to  $x$$\frac{\text{3}}{\text{5}}$(−3) + $\frac{\text{2}}{\text{5}}$(4) which is equivalent to $x$p = −$\frac{\text{1}}{\text{5}}$. Then calculate the $y$-coordinate of P using the weighted averages:  $y$p = $\frac{\text{3}}{\text{5}}$$y$1 + $\frac{\text{2}}{\text{5}}$$y$2 which is equivalent to $y$p = $\frac{\text{3}}{\text{5}}$(6) + $\frac{\text{2}}{\text{5}}$(−8).  Therefore, P is at (−$\frac{\text{1}}{\text{5}}$ , $\frac{\text{2}}{\text{5}}$).
• The second method uses the computations of a fraction of the horizontal and the vertical distance between the endpoints (partial distances). That is, if P is partitioning the segment in the ratio a : b or $\frac{\text{a}}{\text{a+b}}$ of the way from ( $x$1 , $y$1) to ($x$2 , $y$2 then its location is $\frac{\text{a}}{\text{a+b}}$ of the horizontal distance and $\frac{\text{a}}{\text{a+b}}$ of the vertical distance from ($x$1 , $y$1) to ($x$2 , $y$2).
• For example, if the given ratio is 2: 3 and the points are at A(−3, 6) and B(4,−8) discuss with students where P is on its way from A to B. Students should be able to determine the horizontal distance from A to B as $x$2$x$ = 4 − (−3), which is 7. Then, determine the vertical distance 2 as $y$2$y$1 = −8 − 6, which is −14. Since the ratio is 2: 3, P is $\frac{\text{2}}{\text{5}}$ of the way from A to B. Students can calculate $\frac{\text{2}}{\text{5}}$ of the horizontal and the vertical distance, $\frac{\text{2}}{\text{5}}$(7) = $\frac{\text{14}}{\text{5}}$  and $\frac{\text{2}}{\text{5}}$−14) = −$\frac{\text{28}}{\text{5}}$, respectively. Students should realize that they will need to add the partial distances to A. Therefore, the coordinates of P are (−3 + $\frac{\text{14}}{\text{5}}$, 6 − $\frac{\text{28}}{\text{5}}$) = (−$\frac{\text{1}}{\text{5}}$ , $\frac{\text{2}}{\text{5}}$).
• Other methods include the use of formulas. It is important to note that this would require students to memorize formulas and not encourage students to explore all of the concepts.
• Instruction include that understanding that the midpoint of a segment partitions that segment in the ratio 1:1.
• Instruction includes various approaches when finding medians or centroids of triangles. (MTR.2.1, MTR.3.1) Different methods are described below.
• The equation of the line containing a median can be written from the coordinates of the vertex and the coordinates of the midpoint of the opposite side. Solving a system of equations formed by the equations of the two lines containing two medians of a triangle will result in the coordinates of the centroid.
• The centroid formula can be given as Centroid = , where ($x$1 , $y$1),($x$2 , $y$2) and ($x$3 + y3) are the coordinates of the vertices of the triangle. Connections should be made to the fact that the coordinates of the centroid are the means of the coordinates of the vertices.
• The centroid is also the center of gravity of the triangle. Show them with a cardboard triangle that the triangle balances perfectly on its centroid (use a pencil or the tip of your finger).
• A method that makes connections to partitions can be found in the Centroid 2 Theorem. This theorem states that the centroid is$\frac{\text{2}}{\text{3}}$ of the distance from each 3 vertex to the midpoint of the opposite side, that is, the centroid partitions the 2 median of the way from the vertex to the midpoint of the opposite side. This 3 theorem can also be used to find the coordinates of centroid.
• For example, given a triangle with vertices A($x$1 + $y$1), B($x$2 + $y$2) and C($x$3 + $y$3), the midpoints of the sides will be (a1 + b1) , (a2 + b2) and (a3 + b3) for  BC, AC and AB, respectively. Applying the Centroid Theorem to the median LC, the centroid is at  ( $\frac{\text{1}}{\text{3}}$$x$3 + $\frac{\text{2}}{\text{3}}$a3, $\frac{\text{1}}{\text{3}}$$y$3 + $\frac{\text{2}}{\text{3}}$b3). Using Substitution property of equality,  = . This results in Centroid = .

### Common Misconceptions or Errors

• Students may confuse using vertical and horizontal distances when partitioning segments.
• Students may need to be reminded that the segment partition refers to the parts, whose sum is the whole, not a part of a whole.
• For example, a segment partitioned in a 1:4 ratio is actually 5 parts separated into 1 part and 4 parts, not 1 out of 4 parts.

• What are the coordinates of the point that partitions segment AB in the ratio 2: 3?

• Circle A has center located at (2, 2) and contains the point (4,4).
• Part A. Write the equation that describes circle A
• Part B. Write the equation of a line tangent to Circle A at (4,4).
• Part C. Find the equation of a vertical tangent line and of a horizontal tangent line.

• Triangle ABC has two of its the vertices located at (−4,−1) and (3,−3).
• Part A. Triangle ABC has a centroid located at (−1, $\frac{\text{1}}{\text{3}}$). What is the third vertex of the triangle?
• Part B. Determine whether triangle ABC is a right triangle based on its angle measures and side lengths.
• Part C. If triangle ABC is not a right triangle, can you classify what type of triangle ABC is?

### Instructional Items

Instructional Item 1
• Given J(−4,2) and K(2,1), find the coordinates of point M on JK that partitions the segment into the ratio 1: 2.

*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))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.3.AP.3: Use coordinate geometry to solve mathematical geometric problems involving lines, triangles and quadrilaterals.

## Related Resources

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

## Formative Assessments

Partitioning a Segment:

Students are asked to find the coordinates of a point which partitions a segment in a given ratio.

Type: Formative Assessment

Centroid Coordinates:

Students are asked to find the coordinates of the centroid when given the ratio of a directed segment.

Type: Formative Assessment

Proving Slope Criterion for Perpendicular Lines - 2:

Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.

Type: Formative Assessment

Proving Slope Criterion for Perpendicular Lines - 1:

Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.

Type: Formative Assessment

Proving Slope Criterion for Parallel Lines - Two:

Students are asked to prove that two lines with equal slopes are parallel.

Type: Formative Assessment

Proving Slope Criterion for Parallel Lines - One:

Students are asked to prove that two parallel lines have equal slopes.

Type: Formative Assessment

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 whether 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

## Lesson Plans

Space Equations:

In this lesson, students model the orbit of a satellite and the trajectory of a missile with a system of equations. They solve the equations both graphically and algebraically.

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

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

The Seven Circles Water Fountain:

Students will apply concepts related to circles, angles, area, and circumference to a design situation.

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

Partition Point For The Queen:

Students will locate a point that partitions a line segment into a given ratio. Students will use a variety of methods; the activities range from informal student definitions and sketches to tasks using number lines and the coordinate plane.

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

Geometree Thievery:

This geometry lesson focuses on partitioning a segment on a coordinate grid in a non-traditional and interesting format. Students will complete a series of problems to determine which farmers are telling the truth about their harvested "Geometrees."

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

Partitioning a Segment:

In this lesson, students find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Type: Lesson Plan

Observing the Centroid:

Students will construct the medians of a triangle then investigate the intersections of the medians.

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

The Centroid:

Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.

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

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

Partition Me:

Students will learn how to partition a segment. Turn your class into a partitioning party; just BYOGP (Bring your own graph paper).

Type: Lesson Plan

Intersecting Medians and the Resulting Ratios:

This lesson leads students to discover empirically that the distance from each vertex to the intersection of the medians of a triangle is two-thirds of the total length of each median.

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

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

## Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

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.

Unit Squares and Triangles:

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

## MFAS Formative Assessments

Centroid Coordinates:

Students are asked to find the coordinates of the centroid when given the ratio of a directed segment.

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.

Partitioning a Segment:

Students are asked to find the coordinates of a point which partitions a segment in a given ratio.

Proving Slope Criterion for Parallel Lines - One:

Students are asked to prove that two parallel lines have equal slopes.

Proving Slope Criterion for Parallel Lines - Two:

Students are asked to prove that two lines with equal slopes are parallel.

Proving Slope Criterion for Perpendicular Lines - 1:

Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal.

Proving Slope Criterion for Perpendicular Lines - 2:

Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular.

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.

## Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast

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.

Unit Squares and Triangles:

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

## Parent Resources

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

## Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>

Type: Perspectives Video: Professional/Enthusiast