Page 1 - Math: Additional Topics Study Guide for the SAT Exam
General Information
There are three topics that are not covered in the other math questions. You will only see about six questions involving these three areas of math: geometry, trigonometry, and complex numbers. Some of them will be in the calculator-allowed section and for some, you will not be able to use a calculator. Review the concepts here and seek extra practice if you do not feel confident in any of these areas of math.
As you review, please remember that you will have access to basic formulas throughout the math sections of the SAT exam and that more obscure formulas will be provided with items requiring their use. What you will need to know is how and when to use these formulas to arrive at a correct answer.
As you proceed through the math section, you will notice that some of the questions require you to come up with your own answer and fill it in on a grid. Check out these sample questons to see how to do that.
Geometry
Items relating to Geometry on the SAT exam will assume you know the basic geometry concepts from pre-high school math and that you learned how to extend these during geometry courses in high school. Here are some things to review.
Lines
Lines are one-dimensional figures that extend through two points to infinity. Consequently, lines do not have a defined length.
The portion of a line between, and including, two points is called a line segment. Line segments have a definite length that can be determined if enough information is provided.
Consider the line segment \(AC\), which includes point \(B\). If line segment \(AC\) measures \(12\), line segment \(AB\) measures \(3x\) and line segment \(BC\) measures \(4x\), what is the length of \(AB\)?
Notice that because \(B\) lies along \(AC\), the length of \(AB\) combined with the length of \(BC\), yields the length of \(AC\). As a result, an algebraic equation can be generated from the information provided:
\[AB + BC = AC\] \[3x + 4x = 12\] \[7x = 12\] \[x = \frac{12}{7}\]Substitute the known value of \(x\) into the expression for \(AB\):
\[3x = 3 \cdot \frac{12}{7} = \frac{36}{7}\]The midpoint of a line segment is the point along the segment that divides the segment into two equal portions.
Consider segment \(XZ\), which has midpoint \(Y\). If \(XY\) measures \(3x + 1\) and \(YZ\) measures \(x + 2\), what is the length of \(XZ\)?
Because \(Y\) is the midpoint, \(XY\) and \(YZ\) are of equal length. Set their lengths equal to each other to solve for the unknown \(x\) then substitute this value into the sum of \(XY\) and \(YZ\) to determine the length of \(XZ\):
\[XY = YZ\] \[3x + 1 = x + 2\] \[2x = 1\] \[x = \frac{1}{2}\]and
\[XY + YZ = XZ\] \[3 \cdot \frac{1}{2} + 1 + \frac{1}{2} + 2 = XZ\] \[\frac{3}{2} + \frac{1}{2} + 3 = XZ\] \[5 = XZ\]Parallel lines are lines that never intersect because they have the same slope, but different \(x\) and \(y\) intercepts (if graphed on a coordinate plane). Parallel lines that intersect are the same line. Consider why this is.
Perpendicular lines are lines that intersect at a 90-degree angle. The slopes of perpendicular lines are negative reciprocals of each other. For example, if the slope of line \(a\) is \(2\), and line \(b\) is perpendicular to \(a\), line \(b\) has a slope of \(-\frac{1}{2}\).
A transversal is a line that passes through two parallel lines. Because parallel lines are essentially the same line, the angles formed by a transversal and one of the parallel lines are equivalent in measure to the corresponding angles formed on the other parallel line.
Angles
The intersection of lines, line segments, or rays generates angles. Acute angles are less than \(90\) degrees, right angles are exactly \(90\) degrees, and obtuse angles are greater than \(90\) degrees.
In the case of a transversal passing through two parallel lines, \(4\) pairs of corresponding angles are formed that have different relationships with each other. Each corresponding angle pair is congruent. Each of these angles also forms two linear pairs with the angles adjacent and a congruent vertical angle pair. The alternate exterior angles formed are congruent and the alternate interior angles formed are congruent, as well.
Straight angles measure \(180\) degrees. And a collection of angles around a point sum to \(360\) degrees.
Triangles
Triangles are polygons with three vertices at the intersection of three line segments. The interior angles of a triangle add to \(180\) degrees.
Isosceles triangles are triangles with two side lengths of equal measurement and two angles of equal measurement.
Equilateral triangles are those with three side lengths of equal measurement. Equilateral triangles also contain three \(60\)-degree angles.
Right triangles contain two acute angles and a right angle. Their side lengths can be related through the Pythagorean Theorem. There are two special right triangles to become familiar with: \(30-60-90\) and \(45-45-90\) right triangles.
A 30-60-90 triangle has angle measurements of \(30\), \(60\), and \(90\) degrees. The side lengths of these triangles are always in the ratio of \(x, \,x\sqrt{3}, \,2x\), corresponding to the \(30-60-90\) angles.
A 45-45-90 triangle has angle measurements of \(45\), \(45\), and \(90\) degrees. The side lengths of these triangles are always in the ratio of \(x, \,x, \,x\sqrt{2}\), corresponding to the \(45-45-90\) angles.
In order to verify that a proposed triangle is valid, compare the sum of two side lengths with the length of the third side. Two sides of a valid triangle must always be longer than the third side.
Consider the following example: Is a three-sided polygon with side lengths \(4\), \(4\), \(12\) a valid triangle?
Examine the sum of each two side lengths in comparison with the third length to determine whether a valid triangle can be made.
\(4 + 4 > 12\)
\(8 > 12\)
This is not true, so a triangle with the given side lengths is impossible.
Other Polygons
Regular polygons are polygons exhibiting angles and side lengths that are all equal in measure.
Squares are regular polygons because each of its angles measures \(90\) degrees, and all side lengths measure the same length. The perimeter of a square is equal to \(4s\) where \(s\) is the side length. The area of a square is equal to \(s^2\).
A rectangle, another quadrilateral, is only regular when it is also a square. The perimeter of a rectangle is \(2l + 2w\), where \(l\) is the length and \(w\) is the width. The area of a rectangle is \(lw\).
A parallelogram is a quadrilateral with parallel lines composing opposite sides. The properties governing transversals and parallel lines also apply to parallelograms. The perimeter of a parallelogram is found by summing the length of each side. The area of a parallelogram is equal \(bh\), where \(b\) is the base and \(h\) is the height.
A trapezoid is a quadrilateral composed of two parallel lines. Its perimeter is found by summing each of its sides. Its area is found by multiplying the average of its bases with its height, otherwise expressed as:
\[A = \frac{1}{2} \cdot (b_1 + b_2) \cdot h\]A rhombus is a quadrilateral that contains equal side lengths but two pairs of different angle measurements. Its perimeter is the sum of its sides and its area is its base times its height.
Congruency and Similarity of Polygons
Geometric figures sharing the same attributes (such as line segments with the same length, polygons with the same side lengths and angle measurements, etc.) are said to be congruent. Congruency is designated with this symbol, \(\cong\).
Figures and shapes that are not the same size, but have proportional measurements are said to be similar. Similarity is designated with this symbol, \(\sim\).
Circles
A circle is a collection of points equidistant from a point at a center.
Any line segment starting and ending on the circle that passes through the center is known as a diameter. A radius is any line segment starting at the circle’s center and ending on the circle.
Line segments that do not pass through the center of the circle are called chords.
The circumference of a circle is the perimeter, or distance around the circle. It is defined as:
\(C = \pi \cdot d\) or \(C = 2 \cdot \pi \cdot r\), where \(d\) is the diameter and \(r\) is the radius.
An arc is a part of the circumference designated by two or three points.
Arc length is defined mathematically in radians through the formula: \(s = r \cdot \theta\), where \(r\) is the radius, and \(\theta\) is the measurement of the central angle. It is defined in degrees with the formula:
\[s = r \cdot \theta \cdot \frac{\pi}{180}\]A sector is the area included inside of a central angle. It is defined mathematically in radians by the formula: \(A = \frac{\theta}{2} \cdot r^2\), where \(\theta\) is the central angle measurement and \(r\) is the radius. The same area is represented in degrees through the following:
\[A = \frac{\theta}{360} \cdot \pi \cdot r^2\]A central angle is any angle with its side lengths formed from two radii and an arc. An inscribed angle is any angle formed from two chords and an arc.
A tangent line is a line forming a \(90\)-degree angle with a radius of the circle at the point along the circle’s edge.
Geometric Notation
Become familiar with the following notation:
Point A: \(A\)
Line AB: \(\overleftrightarrow{AB}\)
Line segment AB: \(\overline{AB}\)
Angle A: \(\angle A\)
Measure of angle A: \(m\angle A\)
Length of segment AB: \(AB = 5\)
Other Notation
Also, develop familiarity with the following notation:
Ray XY: \(\overrightarrow{XY}\)
Angle ABC (vertex B): \(\angle ABC\)
Triangle ABC: \(\bigtriangleup ABC\)
Quadrilateral XYZW: \(XYZW\)
Line A parallel with line B: \(A \parallel B\)
Line X perpendicular with line Y: \(X \perp Y\)
The Pythagorean Theorem
The Pythagorean Theorem relates the three sides of a right triangle.
The shorter sides of a right triangle are known as the legs and the longest side is known as the hypotenuse. The Pythagorean Theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse:
\[a^2 + b^2 = c^2\]where \(a\) and \(b\) are legs, and \(c\) is the hypotenuse.
Arc and Circle Notation
The arc of a circle is a portion of the circle’s circumferences connecting two points, it is represented with two or three points along the circle with a curved bar above them, for example:
\(\overparen{\rm CH}\) represents arc \(CH\)
Circles are referenced through their center and can be represented by a circle with a point inside of it.
Using Figures “Not Drawn to Scale”
Unless otherwise specified, all figures presented are to be interpreted as not drawn to scale. This is important to keep in mind; as a result, lines and angles that appear to be long or short, big or small, cannot be assumed to be so.
Rather than assuming information based on the visual representation of a figure, it is necessary to build up conclusions about the figure based on the information given about the figure.
It is, however, safe to assume that lines (unless otherwise stated), are indeed straight. All other conclusions should be based upon knowledge of shapes, theorems, and postulates.