New SAT Math Workbook

A line with a slope of 1 slopes upward from left to

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Unformatted text preview: 279 2. TANGENT LINES AND INSCRIBED CIRCLES A circle is tangent to a line (or line segment) if it intersects the line (or line segment) at one and only one point (called the point of tangency). In addition to the rules you learned in Chapter 13 involving tangents, for the new SAT you should know the following two rules: 1. A line (or line segment) that is tangent to a circle is always perpendicular to a radius drawn from the circle’s center to the point of tangency. Thus, in the next figure, which shows a circle with center O, OP 9 AB : 2. For any regular polygon (in which all sides are congruent) that circumscribes a circle, the point of tangency between each line segment and the circle bisects the segment. Thus, in the next figure, which shows three circles, each circumscribed by a regular polygon (shown from left to right, an equilateral triangle, a square, and a regular pentagon), all line segments are bisected by the points of tangency highlighted along the circles’ circumferences: These two additional rules involving tangents allow for a variety of additional types of SAT questions. Example: In the figure below, AB passes through the center of circle O and AC is tangent to the circle at P. If the radius of the circle is 3 and m∠OAC = 30°, what is the area of the shaded region? 3 (A) (3 3 − π ) 2 (B) (C) (D) (E) 1 (3 − 2 3π ) − 3) 4 3−π 2 (3π 3 4π − 3 280 Chapter 16 Solution: The correct answer is (A). Draw a radius from O to P. Since AC is tangent to the circle at P, AC ⊥ PO , and drawing the radius from O to P forms a right triangle (∆AOP), whose area = 1 2 circle, 360), and hence the segment of the circle bound by ∠OAC is one-sixth the circle’s area, or 1 π 32 6 3 = 9 π = 2 π . To answer the question, subtract the area of this segment of the circle from the 6 (3)(3 3= ) 9 2 3 . Since m∠OAP = 30°, m∠OAP = 60° (one sixth the total number of degrees in the area of ∆AOP: 9 2 3 3 3 − 2 π = 2 (3 3 − π ) . Additional Geometry Topi...
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This note was uploaded on 08/15/2010 for the course MATH a4d4 taught by Professor Colon during the Spring '10 term at Embry-Riddle FL/AZ.

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