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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
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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,
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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
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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?
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