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Circles: Definition, Parts, and Properties

Learn about the definition of a circle, its parts, and various properties such as similarity, tangents, and intersecting tangents. Explore how circles can be transformed and modeled in different ways.

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Circles: Definition, Parts, and Properties

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  1. Circles

  2. Information

  3. Definition of a circle A circle is a set of all points in a plane that are equidistant from a given point. This given point is the center of the circle. center This is why a compass is used to draw a circle – the pencil is always held at an equal distance from the center of the circle.

  4. Parts of circle

  5. Circles and similarity Are circles A and B similar? Prove your answer. Circle A can be mapped to circle B by: • a translation (x, y) → (x + 4, y + 6) B • and a dilation with center (8, 11) and scale factor ⅔. A To prove that two shapes are similar, it must be shown that they are related by similarity transformations. So circles A and B are similar.

  6. Circles and similarity proof Are all circles similar? Prove your answer. Any circle with center (x, y) and radius r can be mapped to another circle with center (x + a, y + b) and radius s by: (x + a, y + b) s • a translation (x, y) → (x + a, y + b) • and a dilation with center (x + a, y + b)and scale factor . s r (x, y) The new circle is a dilation and a translation of the original circle. Dilations and translations are both similarity transformations. r So any circle is similar to any other circle.

  7. Lines in a circle

  8. Pairs of circles

  9. Exploring radius and tangent

  10. Radius and tangent If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. A If a line is perpendicular to a radius of a circle at a point on a circle, then the line is tangent to the circle. The first theorem is the converse of the second theorem.

  11. Radius and tangent proof The line, l, lies on the circle A and is perpendicular to the radius at that point. Is l also a tangent to the circle? Choose any point on lthat is not B. Call this point C. We are told that CB is perpendicular to AB. This means that CB and AB form a right angle in △ABC. apply the Pythagorean theorem: C B l AB² + BC² = AC² (1) AB² + BC² > AB² BC > 0: (2) AC² > AB² substituting: A AC > AB simplifying: Therefore AC≠AB Ccannot lie on the circle. The line l intersects with the circle at one point only, meaning that l is a tangent to the circle.

  12. Exploring intersecting tangents

  13. Intersecting tangents If two segments are tangent to a circle from the same external point, then the segments are congruent. The figure below shows two tangents to a circle, P. Use this theorem to find the value of x. by the theorem: AB is congruent to BC. A 8 + x definition of congruence: AB = BC B P substituting values: 8+x = 3x 3x simplifying: 8 = 2x C x = 4

  14. Constructing a tangent line

  15. Modeling with circles

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