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File: Geometry Pdf 167137 | Official Sat Study Guide Additional Topics Math
chapter 19 additional topics in math in addition to the questions in heart of algebra problem solving and remember data analysis and passport to advanced math the sat math test ...

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            CHAPTER 19
            Additional Topics 
            in Math
            In addition to the questions in Heart of Algebra, Problem Solving and         REMEMBER
            Data Analysis, and Passport to Advanced Math, the SAT Math Test               Six of the 58 questions 
            includes  several questions that are drawn from areas of geometry,            (approximately 10%) on the SAT 
            trigonometry, and the arithmetic of complex numbers. They include             Math Test will be drawn from 
            both multiple-choice and student-produced response questions. Some            Additional Topics in Math, which 
            of these questions appear in the no-calculator portion, where the use of      includes geometry, trigonometry, 
            a calculator is not permitted, and others are in the calculator portion,      and the arithmetic of complex 
            where the use of a calculator is permitted.                                   numbers.
            Let’s explore the content and skills assessed by these questions.
            Geometry                                                                      REMEMBER
            The SAT Math Test includes questions that assess your understanding 
            of the key concepts in the geometry of lines, angles, triangles, circles,     You do not need to memorize a large 
            and other geometric objects. Other questions may also ask you to find         collection of geometry formulas. 
            the area, surface area, or volume of an abstract figure or a real-life        Many geometry formulas are 
            object. You don’t need to memorize a large collection of formulas, but        provided on the SAT Math Test in the 
            you should be comfortable understanding and using these formulas to           Reference section of the directions.
            solve various types of problems. Many of the geometry formulas are 
            provided in the reference information at the beginning of each section 
            of the SAT Math Test, and less commonly used formulas required to 
            answer a question are given with the question.
            To answer geometry questions on the SAT Math Test, you should 
            recall the geometry definitions learned prior to high school and know 
            the essential concepts extended while learning geometry in high 
            school. You should also be familiar with basic geometric notation.
            Here are some of the areas that may be the focus of some questions on 
            the SAT Math Test.
              § Lines and angles
                    Lengths and midpoints
                  w 
                    Measures of angles
                  w 
                    Vertical angles
                  w 
                    Angle addition
                  w 
                    Straight angles and the sum of the angles about a point
                  w 
                                                                                                                       241
       PART 3 | Math
                                                           Properties of parallel lines and the angles formed when parallel
                                                         w 
                                                           lines are cut by a transversal
                                                           Properties of perpendicular lines
                                                         w 
                                                     § Triangles and other polygons
                                                           Right triangles and the Pythagorean theorem
                                                         w 
                                                           Properties of equilateral and isosceles triangles
                                                         w 
                                                           Properties of 30°-60°-90° triangles and 45°-45°-90° triangles
                                                         w 
                                                           Congruent triangles and other congruent figures
       PRACTICE AT                                       w 
       satpractice.org                                     Similar triangles and other similar figures
                                                         w 
       The triangle inequality theorem                     The triangle inequality
       states that for any triangle, the                 w 
                                                           Squares, rectangles, parallelograms, trapezoids, and other
       length of any side of the triangle                w 
       must be less than the sum of the                    quadrilaterals
       lengths of the other two sides of                   Regular polygons
       the triangle and greater than the                 w 
       difference of the lengths of the              § Circles
       other two sides.                                    Radius, diameter, and circumference
                                                         w 
                                                           Measure of central angles and inscribed angles
                                                         w 
                                                           Arc length, arc measure, and area of sectors
                                                         w 
                                                           Tangents and chords
                                                         w 
                                                     § Area and volume
                                                           Area of plane figures
                                                         w 
                                                           Volume of solids
                                                         w 
                                                           Surface area of solids
                                                         w 
                                                   You should be familiar with the geometric notation for points and lines, 
                                                   line segments, angles and their measures, and lengths.
                                                                                         y e               m
                                                                                       E
                                                                                  P     4
                                                                                        2           D
                                                                                          B Q
                                                                       M                                      x
                                                                           –4    –2     O       2     4
                                                                                       –2           C
                                                                                      –4
                                                   In the figure above, the xy-plane has origin O. The values of x on the 
                                                   horizontal x-axis increase as you move to the right, and the values of y 
                                                   on the vertical y-axis increase as you move up. Line e contains point P, 
        242
                                                                                                                                                                ChAPTeR 19 | Additional Topics in Math
                      which has coordinates (−2, 3); point E, which has coordinates (0, 5); 
                      and point M, which has coordinates (−5, 0). Line m passes through the 
                      origin O (0, 0), the point Q (1, 1), and the point D (3, 3).
                      Lines e and m are parallel—they never meet. This is written e || m.                                                                           PRACTICE AT
                      You will also need to know the following notation:                                                                                            satpractice.org
                             _                                                                                                                                      Familiarize yourself with these 
                             ‹    ›
                         § PE :   the line containing the points P and E (this is the same as line e )                                                              notations in order to avoid 
                             _                                                                                                                                      confusion on test day.
                            PE   or line segment PE : the line segment with endpoints P and E
                         § 
                                                                                                                             __
                             PE : the length of segment PE (you can write PE = 2   2     )
                         §                                                                                                 √
                             _
                                 ›
                         § PE       : the ray starting at point P and extending indefinitely in the
                             direction of point E
                             _
                                 ›
                         § EP        : the ray starting at point E and extending indefinitely in the
                             direction of point P
                                                                                 _ _
                                                                                      ›              ›
                         § ∠DOC: the angle formed by OD                                   and O  C   
                         § △PEB: the triangle with vertices P, E, and B
                         § Quadrilateral BPMO: the quadrilateral with vertices B, P, M, and O
                             _ _
                         § BP       ⊥  PM :  segment BP is perpendicular to segment PM (you should
                             also recognize that the right angle box within ∠BPM means this
                             angle is a right angle)
                      example 1
                                                A                            12                           D
                                                                                                                               
                                                                                                                  5
                                                                                                        E
                                                                                                  1                            m
                                                                                                          B            C
                         In the figure above, line ℓ is parallel to line m, segment BD is perpendicular to 
                         line m, and segment AC and segment BD intersect at E. What is the length of 
                         segment AC?
                      Since segment AC and segment BD intersect at E, ∠AED and ∠CEB are 
                      vertical angles, and so the measure of ∠AED is equal to the measure of                                                                        PRACTICE AT
                      ∠CEB. Since line ℓ is parallel to line m, ∠BCE and ∠DAE are alternate 
                      interior angles of parallel lines cut by a transversal, and so the measure                                                                    satpractice.org
                      of ∠BCE is equal to the measure of ∠DAE. By the angle-angle theorem,                                                                          A shortcut here is remembering that 
                      △AED is similar to △CEB, with vertices A, E, and D corresponding to                                                                           5, 12, 13 is a Pythagorean triple  
                      vertices C, E, and B, respectively.                                                                                                           (5 and 12 are the lengths of the sides 
                      Also, △AED is a right triangle, so by the Pythagorean theorem,                                                                                of the right triangle, and 13 is the 
                                  ___                                                                                                                               length of the hypotenuse). Another 
                                        2          2             2       2
                      AE =   AD  + DE    =    12  + 5    =    169                         = 13.    Since △AED is similar to                                         common Pythagorean triple is 3, 4, 5.
                               √                          √                    √
                      △CEB, the ratios of the lengths of corresponding sides of the two 
                                                                                                                                                                                                                          243
             PART 3 | Math
                                                                                                                                                                                                        ED 5
                                                                                                                                                                                                        _ _
                                                                                                  triangles are in the same proportion, which is                                                                           = 5. Thus,  
                                                                                                                                                                                                                 =     
                                                                                                                                                                                                        EB          1
                                                                                                  AE           13                                           13                                                                            13          78
                                                                                                  _ _                                                       _                                                                             _ _
                                                                                                            =           = 5, and so EC =      .    Therefore, AC = AE + EC = 13 +         =                                                                . 
                                                                                                                                                              5                                                                             5          5  
                                                                                                   EC          EC
                                                                                                  Note some of the key concepts that were used in Example 1:
                                                                                                      § Vertical angles have the same measure.
                                                                                                      § When parallel lines are cut by a transversal, the alternate interior 
                                                                                                           angles have the same measure.
                                                                                                      § If two angles of a triangle are congruent to (have the same measure 
                                                                                                           as) two angles of another triangle, the two triangles are similar.
             PRACTICE AT                                                                              § The Pythagorean theorem: a2 + b2 = c2, where a and b are the 
             satpractice.org                                                                               lengths of the legs of a right triangle and c is the length of the 
             Note how Example 1 requires the                                                               hypotenuse.
             knowledge and application of                                                             § If two triangles are similar, then all ratios of lengths of 
             numerous fundamental geometry                                                                 corresponding sides are equal.
             concepts. Develop mastery of 
             the fundamental concepts and                                                             § If point E lies on line segment AC, then AC = AE + EC.
             practice applying them on test-like                                                  Note that if two triangles or other polygons are similar or congruent, 
             questions.                                                                           the order in which the vertices are named does not necessarily indicate 
                                                                                                  how the vertices correspond in the similarity or congruence. Thus, it 
                                                                                                  was stated explicitly in Example 1 that “△AED is similar to △CEB, with 
                                                                                                  vertices A, E, and D corresponding to vertices C, E, and B, respectively.”
                                                                                                  You should also be familiar with the symbols for congruence and 
                                                                                                  similarity. 
                                                                                                      § Triangle ABC is congruent to triangle DEF, with vertices A, B, and C 
                                                                                                           corresponding to vertices D, E, and F, respectively, and can be 
                                                                                                           written as △ABC ≅ △DEF. Note that this statement, written with the 
                                                                                                           symbol ≅, indicates that vertices A, B, and C correspond to vertices D, 
                                                                                                           E, and F, respectively. 
                                                                                                      § Triangle ABC is similar to triangle DEF, with vertices A, B, and C 
                                                                                                           corresponding to vertices D, E, and F, respectively, and can be 
                                                                                                           written as △ABC ~ △DEF. Note that this statement, written with 
                                                                                                           the symbol ~, indicates that vertices A, B, and C correspond to 
                                                                                                           vertices D, E, and F, respectively.
                                                                                                  example 2
                                                                                                                                                                                            x°
                                                                                                      In the figure above, a regular polygon with 9 sides has been divided into  
                                                                                                      9 congruent isosceles triangles by line segments drawn from the center of the 
                                                                                                      polygon to its vertices. What is the value of x?
              244
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