Tangents and Circles: Unlocking the Equation

Algebra 2 Grades High School 7:22 Video
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Lesson Description

Learn how to determine the equation of a circle given its center and a tangent line. This lesson explores the relationship between tangents, radii, and the standard form of a circle's equation.

Video Resource

Circle Equation From Tangent

Kevinmathscience

Duration: 7:22
Watch on YouTube

Key Concepts

  • Standard form of a circle's equation: (x - h)² + (y - k)² = r²
  • Tangent to a circle: A line that touches the circle at exactly one point.
  • Radius-Tangent Perpendicularity: A radius drawn to the point of tangency is perpendicular to the tangent line.

Learning Objectives

  • Students will be able to identify the center and radius of a circle given its equation in standard form.
  • Students will be able to determine the equation of a circle given its center and a tangent line.
  • Students will be able to apply the property that a radius is perpendicular to a tangent at the point of tangency.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a circle's equation: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Briefly discuss what a tangent line is and its properties.
  • Video Viewing (10 mins)
    Play the video 'Circle Equation From Tangent' by Kevinmathscience. Encourage students to take notes on the key steps and concepts presented.
  • Guided Practice (15 mins)
    Work through example problems similar to those in the video. Emphasize the importance of visualizing the circle and tangent line to determine the radius. Example 1: Center (2, -3), tangent: y = 1. Example 2: Center (-4, 0), tangent: x = -1.
  • Independent Practice (15 mins)
    Provide students with practice problems to solve independently. Circulate to provide assistance as needed. Problems should vary in difficulty. Example: Center (5, 2), tangent: x-axis. Example: Center (-1, -1), tangent: y-axis.
  • Wrap-up and Discussion (5 mins)
    Review the key concepts and address any remaining questions. Preview the quiz for the next class.

Interactive Exercises

  • GeoGebra Visualization
    Use GeoGebra to create circles and tangent lines. Students can manipulate the center and tangent to observe how the radius changes and how it affects the circle's equation.

Discussion Questions

  • Why is the radius always perpendicular to the tangent at the point of tangency?
  • How does the location of the tangent line relative to the center of the circle help determine the radius?
  • Can you describe a situation where knowing the equation of a circle would be useful in a real-world context?

Skills Developed

  • Problem-solving
  • Visual Reasoning
  • Algebraic Manipulation

Multiple Choice Questions

Question 1:

The standard form of a circle's equation is (x - h)² + (y - k)² = r². What do 'h' and 'k' represent?

Correct Answer: The center's coordinates

Question 2:

A tangent line touches a circle at how many points?

Correct Answer: One

Question 3:

The radius drawn to the point of tangency is always _____ to the tangent line.

Correct Answer: Perpendicular

Question 4:

A circle has a center at (3, -2) and is tangent to the x-axis. What is the radius of the circle?

Correct Answer: 2

Question 5:

A circle has a center at (-1, 4) and is tangent to the y-axis. What is the radius of the circle?

Correct Answer: 1

Question 6:

Which of the following equations represents a circle with center (0,0) and tangent to the line y = 3?

Correct Answer: x² + y² = 9

Question 7:

A circle's equation is (x - 2)² + (y + 1)² = 9. What is the center of the circle?

Correct Answer: (2, -1)

Question 8:

If a circle with center (-3, 5) is tangent to the line y = 5, its radius is:

Correct Answer: Cannot be determined

Question 9:

A circle has its center at (4,-4) and is tangent to the x-axis. Which of the following is the circle's equation?

Correct Answer: (x-4)^2 + (y+4)^2 = 16

Question 10:

What is the radius of a circle with the equation (x+5)^2 + (y-2)^2 = 25?

Correct Answer: 5

Fill in the Blank Questions

Question 1:

The distance from the center of the circle to the tangent line is equal to the circle's ____.

Correct Answer: radius

Question 2:

If a circle's center is at (1, 2) and is tangent to the x-axis, the radius is ____.

Correct Answer: 2

Question 3:

The equation (x - 3)² + (y + 4)² = 16 represents a circle with a center at (____, ____).

Correct Answer: 3,-4

Question 4:

A line that touches a circle at only one point is called a ____.

Correct Answer: tangent

Question 5:

The radius and tangent line at the point of tangency form a ____ degree angle.

Correct Answer: 90

Question 6:

The equation of a circle with a center at the origin and a radius of 7 is x² + y² = ____.

Correct Answer: 49

Question 7:

A circle with a center at (6,-3) is tangent to the y-axis, the radius is ____.

Correct Answer: 6

Question 8:

The general equation for a circle is (x-h)^2 + (y-k)^2 = r^2. In this equation 'r' stands for the ____.

Correct Answer: radius

Question 9:

If a circle has the equation (x-5)^2 + (y+1)^2 = 36, the radius squared, r^2, is equal to ____.

Correct Answer: 36

Question 10:

A circle is tangent to a line. The tangent line intersects the circle at only ____ point(s).

Correct Answer: one

Educational Standards

A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain.

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