Intro to Circles

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Intro to Circles

G.12D

Objective

​TEKS

 (12) Circles. The student uses the process skills to understand geometric relationships and apply theorems and equations about circles. The student is expected to:

    (A) apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems;

    (B) apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems;

    (C) apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems;

    (D) describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle; and

    (E) show that the equation of a circle with center at the origin and radius r is x² + y² = r² and determine the equation for the graph of a circle with radius r and center (h, k), (x - h)² + (y - k)² = r² .

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Open Ended

What are some of the characteristics of a circle?

Vocabulary

​Circle	​
Arc​	Major arc​
Central Angle​	Minor arc​
Chord​	Point of tangency​
Circumference​	Radian measure​
Diameter​	Radius​
Inscribed angle​	Secant​
Intercepted arc​	Tangent segment​

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Segments of a circle

Angles and arcs of a circle

2π (rad) = 360⁰

​Radians and Degrees

x² + y² = r²

​Equation of a circle

Formulas for today

Open Ended

What does the equation of a circle remind you of?

x² + y² = r²

Poll

Who said Pythagorean Theorem?

• I did!!!

• I had no clue...

• I forgot to submit.

How does it work?

​Circles on a coordinate plane

• In order to write the equation of a line, you need the center point and either the radius or a point on the circumference of the circle.

What if...?

​Center point not at origin point

• If the center point is not at origin point we can use:
  (x-h)² + (y-k)² = r² where (h,k) are the coordinates of the center point.

Remember!!!

• (x-h)² moves right

• (y-k)² moves up

Multiple Choice

Where is the center point?

$$\left(x+3\right)^2+\left(y-4\right)^2=16$$

• (3,4)

• (-3,4)

• (4,3)

• (3,-4)

So you have a circle, now what?

​Let's draw some angles!

Using Radians instead of Degrees

Multiple Choice

How many radians are in a 90° angle?

• π

• 2π

• $$\frac{\pi}{4}$$

• $$\frac{\pi}{2}$$

Time to practice!

Partners:

• Work with a partner to draw 4 circles based off of equations found in Canvas

• Yes, it is for a grade.

Independent:

• Work by yourself to complete the quizzes on Canvas:

  • Equations of Circles

  • Converting Radians & Degrees.