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7.04: Honors Activity Part B

Tangible means to touch. Tangent lines are lines that touch a curve at exactly one point. The resulting geometry is a line placed right next to a curve. The model shows a portion of a roller coaster track and the velocities at each point along the track as tangents. Each tangent line is the best straight-line approximation of the curve at that point, and the collection of all the points of tangencies helps to define the curve, which may not be a parabola, circle, ellipse, or hyperbola.
It can be shown that the tangent line and radius of curvature for the corresponding curve are_______________. Let’s take the case of a circle. We will create a line tangent to circle C at point T, resulting in a radius CT.
Therefore, the ____________ of a circle is always ________________ to a tangent at the point of tangency, which in this case is T.

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HONORS ONLY- This lesson is for honors students only. If you are not taking this course for honors credit, you may skip this lesson.

The shortest distance from a point to a line is along a segment perpendicular to the line. The radius is the shortest distance to the tangent line. The radius is perpendicular to the tangent line at point A.

Constructing a Tangent Line 📼 Watch Video, click here Online compass
1.	Start with a circle with it’s center point marked and an exterior point to the circle.
2.	Connect the exterior point with the center of the circle.
3.	Construct the______________ ___________of the segment created.
4.	Set compass width to that of the ______ and the ___________ of the perpendicular lines.
5.	Swing an arc that ____________the circle in two places.
6.	Connect the exterior point to the intersection of the arcs and the circle. You will end up with two tangent lines.

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On their current project, Maurice and Johanna need to create a walkway that goes from the back door of a house straight to a circular pond in the backyard. This will call for the construction of a tangent line.

7.04

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If you ____________ a circle, it will be congruent to the original circle, just in a different location. If a __________ is performed, the dilated circle will be _____________ to the original circle. Given a pair of _____________circles, a dilation from the common center of the circles has a scale factor of k = ____________ Key Idea: All circles are similar! The circles can be proven similar if one or more transformations can be found that map one figure onto another.	r1 is the preimage (the original radius) r2 is the image (the radius of the circle after the dilation)

Explain how circle A with the center at (2, 3) and a radius 1 is similar to circle B with the center at (5, 6) and a radius 4. Translation rule _______________ Scale factor _________________

Explain how circle C with center at (−9, 8) and radius 7 is similar to circle D with center at (−1, −1) and radius 5 Translation rule _______________ Scale factor _________________
Two transformations are performed on circle C to show that circle C is similar to circle A. First, circle C is dilated from the center C, and then it is translated. What are the algebraic descriptions of the transformations?

7.04

ESSENTIAL QUESTIONS - Write out your thoughts!