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CHAPTER 19
Additional Topics
in Math
In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are drawn from areas of geometry, trigonometry, and the arithmetic of complex numbers. They include both multiple-choice and student-produced response questions. Some of these questions appear in the no-calculator portion, where the use of a calculator is not permitted, and others are in the calculator portion, where the use of a calculator is permitted.
Let’s explore the content and skills assessed by these questions.
REMEMBER
Six of the 58 questions (approximately 10%) on the SAT Math Test will be drawn from Additional Topics in Math, which includes geometry, trigonometry, and the arithmetic of complex numbers.
Geometry
The SAT Math Test includes questions that assess your understanding of the key concepts in the geometry of lines, angles, triangles, circles, and other geometric objects. Other questions may also ask you to find the area, surface area, or volume of an abstract figure or a real-life object. You don’t need to memorize a large collection of formulas, but you should be comfortable understanding and using these formulas to 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.
REMEMBER
You do not need to memorize a large collection of geometry formulas. Many geometry formulas are provided on the SAT Math Test in the Reference section of the directions.
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
w 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
PART 3 | Math
w Properties of parallel lines and the angles formed when parallel
lines are cut by a transversal
w Properties of perpendicular lines
§ Triangles and other polygons
w 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
w Similar triangles and other similar figures
w The triangle inequality
w Squares, rectangles, parallelograms, trapezoids, and other
quadrilaterals
w Regular polygons
§ Circles
w 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
§ Area and volume
w Area of plane figures
w Volume of solids
w Surface area of solids
PRACTICE AT
satpractice.org The triangle inequality theorem states that for any triangle, the length of any side of the triangle must be less than the sum of the lengths of the other two sides of the triangle and greater than the difference of the lengths of the other two sides.
You should be familiar with the geometric notation for points and lines, line segments, angles and their measures, and lengths.
O x
e m
y
P
E
B Q
D
C
M
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 ,
PART 3 | Math
triangles are in the same proportion, which is^ ED _ EB =^5 _ 1 = 5. Thus, _^ AE EC =
_^13
EC = 5,^ and so^ EC^ =
13 _
5.^ Therefore,^ AC^ =^ AE^ +^ EC^ = 13 +
13 _
78 _
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.
§ The Pythagorean theorem: a^2 + b^2 = c^2 , where a and b are the
lengths of the legs of a right triangle and c is the length of the hypotenuse.
§ If two triangles are similar, then all ratios of lengths of
corresponding sides are equal.
§ If point E lies on line segment AC , then AC = AE + EC.
Note that if two triangles or other polygons are similar or congruent, 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.
PRACTICE AT
satpractice.org Note how Example 1 requires the knowledge and application of numerous fundamental geometry concepts. Develop mastery of the fundamental concepts and practice applying them on test-like questions.
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?
ChAPTeR 19 | Additional Topics in Math
The sum of the measures of the angles around a point is 360°. Since the 9 triangles are congruent, the measures of each of the 9 angles are
equal. Thus, the measure of each of the 9 angles around the center
point is 360°_ 9 = 40°. In any triangle, the sum of the measures of the
interior angles is 180°. So in each triangle, the sum of the measures of the remaining two angles is 180° − 40° = 140°. Since each triangle is isosceles, the measure of each of these two angles is the same.
Therefore, the measure of each of these angles is 140°_ 2 = 70°. Hence,
the value of x is 70.
Note some of the key concepts that were used in Example 2:
§ The sum of the measures of the angles about a point is 360°.
§ Corresponding angles of congruent triangles have the same
measure.
§ The sum of the measure of the interior angles of any triangle is 180°.
§ In an isosceles triangle, the angles opposite the sides of equal
length are of equal measure.
example 3
B
A
X
Y
In the figure above, ∠AXB and ∠AYB are inscribed in the circle. Which of the following statements is true? A) The measure of ∠AXB is greater than the measure of ∠AYB. B) The measure of ∠AXB is less than the measure of ∠AYB. C) The measure of ∠AXB is equal to the measure of ∠AYB. D) There is not enough information to determine the relationship between the measure of ∠AXB and the measure of ∠AYB.
Choice C is correct. Let the measure of arc AB ^ be d °. Since ∠ AXB is inscribed in the circle and intercepts arc AB ^ , the measure of ∠ AXB is
equal to half the measure of arc AB .^ Thus, the measure of ∠ AXB is d _ 2^ °.
Similarly, since ∠ AYB is also inscribed in the circle and intercepts
arc AB ,^ the measure of ∠ AYB is also d _ 2^ °. Therefore, the measure of
∠ AXB is equal to the measure of ∠ AYB.
Note the key concept that was used in Example 3:
§ The measure of an angle inscribed in a circle is equal to half the
measure of its intercepted arc.
PRACTICE AT
satpractice.org At first glance, it may appear as though there's not enough information to determine the relationship between the two angle measures. One key to this question is identifying what is the same about the two angle measures. In this
case, both angles intercept arc AB .
ChAPTeR 19 | Additional Topics in Math
§ The length of AD
_ is less than the length of
_ AC.
§ The measure of angle ABD is less than the measure of angle ABC.
You may not assume the following from the figure: _ _
§ The length of AD is l ess than the length of DC.
§ The measures of angles BAD and DBA are equal.
§ The measure of angle DBC is greater than the measure of angle ABD.
§ Angle DBC is a right angle.
example 4
O
B
A C
In the given figure, O is the center of the circle, segment BC is tangent to the circle at B, and A lies on segment OC. If OB = AC = 6, what is the area of the shaded region? A) 18√3 − 3π
__
B) 18√3 − 6π
__
C) 36√3 − 3π
__
D) 36√3 − 6π
__
Since segment BC is tangent to the circle at B , it follows that BC ⊥ OB ,
and so triangle OBC is a right triangle with its right angle at B. Since OB = 6 and OB and OA are both radii of the circle, OA = OB = 6, and OC = OA + AC = 12. Thus, triangle OBC is a right triangle with the length of the hypotenuse ( OC = 12) twice the length of one of its legs ( OB = 6). It follows that triangle OBC is a 30°-60°-90° triangle with its 30° angle at C and its 60° angle at O. The area of the shaded region is the area of triangle OBC minus the area of the sector bounded by radii OA and OB.
PRACTICE AT
satpractice.org On complex multistep questions such as Example 4, start by identifying the task (finding the area of the shaded region) and considering the intermediate steps that you’ll need to solve for (the area of triangle OBC and the area of sector OBA ) in order to get to the final answer. Breaking up this question into a series of smaller questions will make it more manageable.
In the 30°-60°-90° triangle OBC , the length of side OB , which is opposite the 30° angle, is 6. Thus, the length of side BC , which is opposite the 60° angle, is (^6) √ 3
_
. Hence, the area of triangle OBC
is _^12 (6)(6√3) = 18√.
_ _ Since the sector bounded by radii OA and OB
has central angle 60°, the area of this sector is _ 360 60 =^1 _ 6 of the area of
the circle. Since the circle has radius 6, its area is π (6) 2 = 36 π , and so
the area of the sector is 1 _ 6 (36 π ) = 6 π. Therefore, the area of the shaded
region is (^18) √ 3 − 6 π ,
_ which is choice B.
PART 3 | Math
Note some of the key concepts that were used in Example 4:
§ A tangent to a circle is perpendicular to the radius of the circle
drawn to the point of tangency.
§ Properties of 30°-60°-90° triangles.
§ Area of a circle.
§ The area of a sector with central angle x ° is equal to — 360^ x^ of the
area of the entire circle.
PRACTICE AT
satpractice.org Arc length, area of a sector, and central angle are all proportional to each other in a circle. This proportionality is written as __^ arc length circumference =^
central angle__ 360 degrees
= area of a sector__area of a circle example 5
Z
X
W
a
b Y
Trapezoid WXYZ is shown above. How much greater is the area of this trapezoid than the area of a parallelogram with side lengths a and b and base angles of measure 45° and 135°? A) 1 _ 2 a^2 B) √2a^2
_
C) 1 _ 2 ab D) √2ab
_
PRACTICE AT
satpractice.org
Note how drawing the parallelogram within trapezoid WXYZ makes it much easier to compare the areas of the two shapes, minimizing the amount of calculation needed to arrive at the solution. Be on the lookout for time-saving shortcuts such as this one.
In the figure, draw a line segment from Y to the point P on side WZ
of the trapezoid such that ∠ YPW has measure 135°, as shown in the
figure below.
P
Z
X
W
a
b Y
Since in trapezoid WXYZ side XY is parallel to side WZ , it follows that WXYP is a parallelogram with side lengths a and b and base angles of measure 45° and 135°. Thus, the area of the trapezoid is greater than a parallelogram with side lengths a and b and base angles of measure
45° and 135° by the area of triangle PYZ. Since ∠ YPW has measure
135°, it follows that ∠ YPZ has measure 45°. Hence, triangle PYZ is a
45°-45°-90° triangle with legs of length a. Therefore, its area is 1 _ 2 a^2 , which is choice A. Note some of the key concepts that were used in Example 5:
§ Properties of trapezoids and parallelograms
§ Area of a 45°-45°-90° triangle
PART 3 | Math
figure in the coordinate plane. You should know that the graph of ( x − a ) 2 + ( y − b ) 2 = r^2 in the xy -plane is a circle with center ( a , b ) and radius r.
PRACTICE AT
satpractice.org You should know that the graph of ( x − a ) 2 + ( y − b ) 2 = r^2 in the xy -plane is a circle with center ( a , b ) and radius r. You should also be comfortable finding the center or radius of a circle from an equation not written in “standard form” by using the method of completing the square to rewrite the equation in standard form.
example 7
x 2 + (y + 1)^2 = 4 The graph of the given equation in the xy-plane is a circle. If the center of this circle is translated 1 unit up and the radius is increased by 1, which of the following is an equation of the resulting circle? A) x 2 + y 2 = 5 B) x 2 + y 2 = 9 C) x^2 + (y + 2)^2 = 5 D) x^2 + (y + 2)^2 = 9
The graph of the equation x^2 + ( y + 1) 2 = 4 in the xy -plane is a circle with center (0, −1) and radius (^) √4 = 2.
_ If the center is translated 1 unit up, the center of the new circle will be (0, 0). If the radius is increased by 1, the radius of the new circle will be 3. Therefore, an equation of the new circle in the xy -plane is x^2 + y^2 = 3^2 = 9, so choice B is correct.
example 8
x 2 + 8x + y 2 − 6y = 24 The graph of the equation above in the xy-plane is a circle. What is the radius of the circle?
The given equation is not in the standard form ( x − a ) 2 + ( y − b ) 2 = r^2. You can put it in standard form by completing the square. Since the coefficient of x is 8 and the coefficient of y is −6, you can write the equation in terms of ( x + 4) 2 and ( y − 3) 2 as follows: x^2 + 8 x + y^2 − 6 y = 24 ( x^2 + 8 x + 16) − 16 + ( y^2 − 6 y + 9) − 9 = 24 ( x + 4) 2 − 16 + ( y − 3) 2 − 9 = 24 ( x + 4) 2 + ( y − 3) 2 = 24 + 16 + 9 ( x + 4) 2 + ( y − 3) 2 = 49 Since 49 = 7^2 , the radius of the circle is 7. (Also, the center of the circle is (−4, 3).)
Trigonometry and Radians Questions on the SAT Math Test may ask you to apply the definitions of right triangle trigonometry. You should also know the definition of radian measure; you may also need to convert between angle measure in degrees and radians. You may need to evaluate trigonometric functions at benchmark angle measures such as 0, _ π 6^ , _ π 4^ , _ π 3 ,
ChAPTeR 19 | Additional Topics in Math
and π _ 2 radians (which are equal to the angle measures 0°, 30°, 45°, 60°, and 90°, respectively). You will not be asked for values of trigonometric functions that require a calculator.
For an acute angle, the trigonometric functions sine, cosine, and tangent can be defined using right triangles. (Note that the functions are often abbreviated as sin, cos, and tan, respectively.)
A C
B
For ∠ C in the right triangle above:
§ sin(∠ C ) = AB _ BC = ___ length of hypotenuse
length of leg opposite ∠ C
§ cos(∠ C ) = AC _ BC =
length of leg adjacent to _____ ∠ C length of hypotenuse
§ tan(∠ C ) = AB _ AC =
_____ length of leg opposite^ ∠ C length of leg adjacent to ∠ C =
sin(_∠ C ) cos(∠ C )
PRACTICE AT
satpractice.org The acronym “SOHCAHTOA” may help you remember how to compute sine, cosine, and tangent. SOH stands for Sine equals Opposite over Hypotenuse, CAH stands for Cosine equals Adjacent over Hypotenuse, and TOA stands for Tangent equals Opposite over Adjacent.
The functions will often be written as sin C , cos C , and tan C , respectively.
Note that the trigonometric functions are actually functions of the measures of an angle, not the angle itself. Thus, if the measure of ∠ C is, say, 30°, you can write sin(30°), cos(30°), and tan(30°), respectively.
Also note that sine and cosine are cofunctions and that
sin B = length of leg opposite _____ ∠ B length of hypotenuse =^
AC _
BC = cos^ C.^ This is the
complementary angle relationship: sin( x °) = cos(90° − x °).
example 9
Q Y X
Z
R
P
In the figure above, right triangle PQR is similar to right triangle XYZ, with vertices P, Q, and R corresponding to vertices X, Y, and Z, respectively. If cos R = 0.263, what is the value of cos Z?
By the definition of cosine, cos R = _ RPRQ and cos Z = (^) ZXZY _. Since triangle
PQR is similar to triangle XYZ , with vertices P , Q , and R corresponding
to vertices X , Y , and Z , respectively, the ratios (^) RRQ^ _ P and (^) ZXZY _are equal.
Therefore, since cos R = RQ _ RP = 0.263, it follows that cos Z = (^) ZXZY _ = 0.263.
Note that this is why, to find the values of the trigonometric functions of, say, d °, you can use any right triangle with an acute angle of measure d ° and then take the appropriate ratio of lengths of sides.
ChAPTeR 19 | Additional Topics in Math
Also note that since a rotation of 2 π about point O brings you back to the same point on the unit circle, sin( s + 2 π ) = sin( s ), cos( s + 2 π ) = cos( s ), and tan( s + 2 π ) = tan( s ), for any radian measure s.
Let angle DEF be a central angle in a circle of radius r , as shown in the following figure.
D
r
s
F
E
A circle of radius r is similar to a circle of radius 1, with constant of proportionality equal to r. Thus, the length s of the arc intercepted by angle DEF is r times the length of the arc that would be intercepted by an angle of the same measure in a circle of radius 1. Therefore, in the figure above, s = r × (radian measure of angle DEF ), or radian measure of
angle DEF = _ s r.
example 10
O
B
A
y
x
In the figure above, the coordinates of point B are (−√2, √2)
_ _
. What is the measure, in radians, of angle AOB? A) π_ 4 B) π_ 2 C) 3 _ 4 π D) 5 _ 4 π
PRACTICE AT
satpractice.org Always be on the lookout for special right triangles. Here, noticing that segment OB is the hypotenuse of a 45°-45°-90° triangle makes this question easier to solve.
Let C be the point (− (^) √2, 0)
_
. Then triangle BOC , shown in the figure below, is a right triangle with both legs of length (^) √ 2
_ .
O
B
C
A
y
x
PART 3 | Math
Hence, triangle BOC is a 45°-45°-90° triangle. Thus, angle COB has measure 45°, and angle AOB has measure 180° − 45° = 135°. Therefore, the measure of angle AOB in radians is 135° × _360°^2 π = _^34 π , which is choice C.
example 11
sin(x) = cos(K − x) In the equation above, the angle measures are in radians and K is a constant. Which of the following could be the value of K? A) 0 B) π_ 4
C) π^ _ 2
D) π
The complementary angle relationship for sine and cosine implies that the equation sin( x ) = cos( K − x ) holds if K = 90°. Since 90° = _^2 π 360° × 90° =^
_ π 2 radians, the value of^ K^ could be^
_ π 2 ,^ which is choice C.
Complex Numbers The SAT Math Test includes questions on the arithmetic of complex numbers.
REMEMBER
The number i is defined to be the solution to equation_ x^2 = −1. Thus,
i^2 = −1, and i = √−1. The square of any real number is nonnegative. The number^ i^ is defined
to be the solution to the equation x^2 = −1. That is, i^2 = −1, or i = √−1.
_
Note that i^3 = i^2 ( i ) = − i and i^4 = i^2 ( i^2 ) = −1(−1) = 1. A complex number is a number of the form a + bi , where a and b are real number constants and i = √−1.
_ This is called the standard form of a complex number. The number a is called the real part of a + bi , and the number b is called the imaginary part of a + bi. Addition and subtraction of complex numbers are performed by adding their real and complex parts. For example,
§ (−3 − 2 i ) + (4 − i ) = (−3 + 4) + (−2 i + (− i )) = 1 − 3 i
§ (−3 − 2 i ) − (4 − i ) = (−3 − 4) + (−2 i − (− i )) = −7 − i
Multiplication of complex numbers is performed similarly to multiplication of binomials, using the fact that i^2 = −1. For example, (−3 − 2 i )(4 − i ) = (−3)(4) + (−3)(− i ) + (−2 i )(4) + (−2 i )(− i ) = −12 + 3 i − 8 i + (−2)(−1) i^2 = −12 − 5 i + 2 i^2 = −12 – 5 i + 2(−1) = −14 − 5 i
REMEMBER
If you have little experience working with complex numbers, practice adding, subtracting, multiplying, and dividing complex numbers until you are comfortable doing so. You may see complex numbers on the SAT Math Test.
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