Lesson 4 | Polygons and Algebraic Relationships | 10th Grade Mathematics

Objective

Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Expressing Geometric Properties with Equations

G.GPE.B.5 — Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

8.EE.B.6

Expressions and Equations

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.G.B.6

Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Identify right triangles on the coordinate plane using the Pythagorean Theorem.
Describe the right angle as being formed by two line segments that are perpendicular.
Compare equations of the legs of a right triangle and develop a conjecture about the criteria for perpendicular lines.
Apply the knowledge of slope criteria to write equations of the lines that are perpendicular to a given line through a point.
Identify equations of perpendicular lines without graphing.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In this lesson, we are preparing for identification of quadrilaterals based on perpendicular and parallel criteria. This goes back to a fourth-grade geometry standard: 4.G.2 “Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines.” We will not get into more properties of parallelograms specifically or quadrilaterals generally in this lesson.
Students may need to review slope and writing equations of lines before they can fully access this lesson. It is recommended to spend time outside of class building this specific skill.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Is the triangle below a right triangle? Explain your reasoning.

Guiding Questions

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Problem 2

Without graphing, determine whether the two sets of equations are perpendicular. Explain your reasoning.

a. $${y=3x;\space x+3y=4}$$

b. $${y-1=-3(x+2);\space12x+4y=32}$$

Guiding Questions

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Problem 3

Write three equations that are perpendicular to $${\overleftrightarrow{AB}}$$.

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Write the equation of the line that passes through point $${(6,8)}$$ and is perpendicular to the line $${3x+2y=8}$$.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students must write equations for lines that are perpendicular to a given line and pass through a given point.
Include problems such as “Is the triangle $${PQR}$$, where $${P(-7,3)}$$, $${Q(-4,7)}$$, and $${R(1,-3)}$$, a right triangle? If so, which angle is the right angle? Justify your reasoning.”

Open Middle Perpendicular Lines and Slope
EngageNY Mathematics Geometry > Module 4 > Topic B > Lesson 5 — (Focus on the perpendicular line questions)
Open Middle Creating Right Triangles — (Show that the triangle you have created is in fact a right triangle)
Open Middle Perpendicular Lines Through a Given Point
Open Middle Equations of Perpendicular Lines
EngageNY Mathematics Geometry > Module 4 > Topic B > Lesson 6 — (Focus on the perpendicular line questions)
Mathematics Vision Project: Secondary Mathematics One Module 8: Connecting Algebra and Geometry — Lesson 8.2 (Focus on the perpendicular line questions)
Open Middle Finding the Length of a Right Triangle's Altitude

Lesson 3

Lesson 5

Lesson Map

Topic A: Distance on the Coordinate Plane

Use the Pythagorean Theorem to calculate distance on a coordinate plane. Develop a formula for calculating distances.

G.GPE.B.7

Partition horizontal and vertical line segments into equal proportions on a number line.

G.GPE.B.6

Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.

G.GPE.B.6

Topic B: Classify Polygons using Slope Criteria and Proportional Line Segments

Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.

G.GPE.B.5 G.SRT.C.8

Describe and apply the slope criteria for parallel lines.

G.GPE.B.5

Identify and create parallelograms, rectangles, rhombuses, and squares on the coordinate plane.

G.GPE.B.4

Algebraically verify midsegment, median, and parallel line relationships in triangles.

G.CO.C.10 G.SRT.B.4

Algebraically verify diagonal relationships in quadrilaterals and parallelograms.

G.CO.C.11 G.GPE.B.4

Topic C: Area and Perimeter On and Off the Coordinate Plane

Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.

G.GPE.B.7

Calculate and justify the area and perimeter of parallelograms and triangles on the coordinate plane.

G.GPE.B.7

Calculate and justify composite and irregular areas on the coordinate plane.

G.GPE.B.7 N.Q.A.3

Describe how the area changes when a figure is scaled.

G.GPE.B.7

Solve area applications by creating and solving quadratic equations.

A.CED.A.1

Describe a polygon on the coordinate plane using a system of inequalities.

A.CED.A.3 G.CO.A.2

Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities.

A.REI.D.12 G.GPE.B.4 G.GPE.B.7

Polygons and Algebraic Relationships

Objective

Common Core Standards

Core Standards

Expressing Geometric Properties with Equations

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Expressions and Equations

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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