Victory for the ACT Exam 16e TG Sample

514 • M ATH

PROPERTIES OF TANGENT LINES

A line tangent to a circle is perpendicular to the radius drawn to the point of tangency, as illustrated in the following �igure:

STUDENT TEXT, p. 187 16. In the �igure below, O is the center of the circle, and PQ is tangent to the circle at P . If the radius of circle O has a length of 6, what is the area of the shaded portion of the �igure? F. r G. 3 0 r H. 6 r J. 9 0 r

EXPLANATIONS

16. (H) Mathematics/Geometry/ Properties of Tangent Lines and

Triangles/30° − 60° − 90° Triangles and Circles CC: HSG-C.B.5 CCRS: EXP.G.1 Difficulty Level = 4; Teaching Time = 7–10 minutes; Purpose = Example Since the radius is drawn to PQ at the p 30 oi ° nt of tangency, the triangle is a − 60° − 90° triangle, so O 60 + = c . Therefore, the shaded part of the �igure is 360 60 6 1 = of the area of the circle. Since area r circle 2 = r , the area of the shaded portion of the �igure is 6 r : 6 1 6 . 6 2 2 ^ h As tlrt ae trengayt i cvaenl y b, teh ae p“pe ll ii emdi. nSai tnec-ea nt hdegaureesas ”o f the circle is r 6 2 2 ^ h 36 r , and the st hh ea dc ei rdc laer, et ah ei sajruesat ol ef stsh tehsahna od ne de - pf oour rt ti oh no f must be less than one-fourth of 36 r : area 9 < r shaded portion . Therefore, (J) can bb ee ceal ui ms ei ni ta ti se ds .l i(gHh)t lwy ol euslsd tbhea na (gaonodd cgl uo es es ss t to) 09 r .