Consider a circle with radius=10 units and center C(0,0). Point A(6, 8) and Point Q(-8, -6) exist on the circle.
Draw the radius to Point A.
Draw the vertical line segment \overline{AB} to the x-axis.
Draw the horizontal line segment \overline{BC} to the y-axis.
You have just created right triangle \triangle ABC. Repeat these steps to create right triangle \triangle QPC.
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Question 2
2.
Use the Pythagorean Theorem to relate the three side lengths of the right triangles.
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Question 3
3.
The coordinates from A(6,8) satisfy the Pythagorean Theorem:
The coordinates from Q(-8,-6), although they are negative, also satisfy the Pythagorean Theorem:
With this information in mind, what Standard Equation can you write to represent the equation of a circle with center (0,0) and radius r?
Any point with coordinates that satisfy this Standard Equation exists ON the circle. However, it is possible to determine if a point exists INSIDE the circle or OUTSIDE the circle as well!
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Question 4
4.
Given the equation:
conjecture if the points L(4,-2), M(0, -8), and N(8, -7) exist INSIDE the circle, ON the circle, or OUTSIDE the circle.
(You may graph within the Show Your Work space if you would find it helpful!)
Draggable item
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Corresponding Item
L(4, -2)
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INSIDE the circle
M(0, -8)
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ON the circle
N(8, -7)
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OUTSIDE the circle
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Question 5
5.
What is the radius of the circle in each equation?
Draggable item
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Corresponding Item
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The circle below is called the Unit Circle, because it has radius=1 unit and center (0,0).
In this investigation, you will learn how to transform the Unit Circle, both graphically and symbolically.
Unit Circle Equation:
Since ALL circles are similar, EVERY circle is a transformation of the Unit Circle. Both equations below represent a "scaled up" version of the Unit Circle (scale factor of 5). Notice the Standard Form is set equal to r^2. The Transformation Form is set equal to 1, and the scale factor appears in the denominator beneath x and y.
Standard Form:
Transformation Form:
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Question 6
6.
Draw the circle that is represented by this equation in Transformation Form:
A circle can also be transformed by translation. Both equations below represent a "scaled up" version of the Unit Circle (scale factor of 4) that has also been translated 3 units left and 5 units up.
Standard Form:
Transformation Form:
IMPORTANT: The operations exist INSIDE the ( ), so we perform the INVERSE transformation of what may seem to make sense! The new center is NOT (+3, -5) as you may suspect! The new center is actually (-3, +5).
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Question 7
7.
Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.
Equation:
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Question 8
8.
Transform the Unit Circle according to the given equation. Graph the circle on the coordinate plane.
Equation:
But what happens when the horizontal scale change and the vertical scale change have DIFFERENT scale factors? In this situation, a new shape is formed. The circle is transformed into an ellipse!