ALEX Classroom Resource

  

Geometry Module 5, Topic D: Equations for Circles and Their Tangents

  Classroom Resource Information  

Title:

 Geometry Module 5, Topic D: Equations for Circles and Their Tangents

URL:

 https://www.engageny.org/resource/geometry-module-5-topic-d-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

Module 5, Topic D brings in coordinate geometry to establish the equation of a circle. Students solve problems to find the equations of specific tangent lines or the coordinates of specific points of contact. They also express circles via analytic equations.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
6. Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

a. Given the endpoints of the diameter of a circle, use the midpoint formula to find its center and then use the Pythagorean Theorem to find its equation.

b. Derive the distance formula from the Pythagorean Theorem.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given the center (h,k) and radius (r) of a circle,
  • Explain and justify that every point on the circle is a combination of a horizontal and vertical shift from the center with a length equal to the radius.
  • Create a right triangle from the center of a circle to a general point on the circle, and show that the legs of the right triangle are the absolute values of x-h and y-k, and the hypotenuse is r, then apply Pythagorean theorem to show that r2 = (x - h)2 + (y - k)2.


Given the endpoint of the diameter of the circle,
  • Find the center of the circle using the midpoint formula, and write the equation of the circle in standard form using the Pythagorean Theorem.
  • Analyze distance in the coordinate plane and use distance to relate points and lines.
  • Calculate the distance between two points using the Pythagorean Theorem.
  • Generalize methods for determining the distance between two coordinate points.
  • Derive the distance formula using a right triangle and the Pythagorean Theorem.
Teacher Vocabulary:
  • Pythagorean theorem
  • Radius
  • Translation
Knowledge:
Students know:
  • Key features of a circle.
  • The Pythagorean Theorem, Midpoint Formula, Distance Formula.
Skills:
Students are able to:
  • Create a right triangle in a circle using the horizontal and vertical shifts from the center as the legs and the radius of the circle as the hypotenuse.
  • Write the equation of the circle in standard form when given the endpoints of the diameter of a circle, using the midpoint formula to find the circle's center, and then use the Pythagorean Theorem to find the equation of the circle.
  • Find the distance between two points when using the Pythagorean Theorem and use that process to create the Distance Formula.
Understanding:
Students understand that:
  • Circles represent a fixed distance in all directions in a plane from a given point, and a right triangle may be created to show the relationship of the horizontal and vertical shift to the distance,
  • Circles written in standard form are useful for recognizing the center and radius of a circle.
  • The distance formula and Pythagorean Theorem can both be used to find length measurements of segments (or sides of a geometric figure)
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.6.1: Define radius, diameter, midpoint and Pythagorean Theorem.
GEO.6.2: Apply the Pythagorean Theorem to find the distance from the center to a point on the circle.
GEO.6.3: Derive the equation of a circle given the center and the radius.
GEO.6.4: Use the midpoint formula to find the center of a circle based on the endpoints of the diameter.

Prior Knowledge Skills:
  • Identify parts of a circle.
  • Recall how to find circumference of a circle.
  • Recall the meaning of a radius and diameter.
  • Identify all types of angles.
  • Recognize the attributes of a circle.
  • Identify and label parts of a circle.
  • Define diameter, radius, circumference, area of a circle, and formula.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.9.6 Using real-world models (Ex. Pizza or Pie) on a coordinate grid, determine the length of the radius.

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
32. Use coordinates to prove simple geometric theorems algebraically.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given coordinates and geometric theorems and statements defined on a coordinate system, use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others.
Teacher Vocabulary:
  • Simple geometric theorems
  • Simple geometric figures
Knowledge:
Students know:
  • Relationships (e.g. distance, slope of line) between sets of points.
  • Properties of geometric shapes.
  • Coordinate graphing rules and techniques.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Accurately determine what information is needed to prove or disprove a statement or theorem.
  • Accurately find the needed information and explain and justify conclusions.
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
  • Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem.
  • Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.32.1: Apply formulas, and properties of polygons, angles, and lines to draw conclusions from the given information.
GEO.32.2: Identify properties of perpendicular and parallel lines, properties of polygons.
GEO.32.3: Illustrate polygons created by given coordinates on a coordinate plane.
GEO.32.4: Identify distance formula, circle formula, Pythagorean Theorem, midpoint.

Prior Knowledge Skills:
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate an understanding of an extended coordinate plane.
  • Draw and label a 4 quadrant coordinate plane.
  • Draw and extend vertical and horizontal number lines.
  • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
  • Recall how to graph points in all four quadrants of the coordinate plane.
  • Define ordered pairs.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Calculate the perimeter and area using the distance between the vertices.
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Recognize examples of right triangles.
  • Demonstrate how to find square roots.
  • Solve problems with exponents.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Tags: algebra, center, circle, coordinates, equation, geometric theorems, Pythagorean Theorem, radius, square
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Comments

There are three lessons on this topic.

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Author: Hannah Bradley
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