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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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