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This chapter is from the book

This chapter is from the book

Geometry

This book assumes a basic understanding of both coordinate geometry and plane geometry. The focus will be on reviewing some general concepts and applying those concepts to questions that might appear on the PSAT math sections.

Understanding the Coordinate Plane

The xy-coordinate plane has four separate quadrants, as shown in Figure 3.1.

Figure 3.1

Figure 3.1 The xy-coordinate plane.

The x-coordinates in Quadrants I and IV will be positive, and the x-coordinates in Quadrants II and III will be negative. The y-coordinates in Quadrants I and II will be positive, and the y-coordinates in Quadrants III and IV will be negative.

Understanding the Equation of a Line

The PSAT will include questions concerning the slope-intercept form of a line, which is expressed as y = mx + b, where m is the slope of the line and b is the y-intercept (that is, the point at which the graph of the line crosses the y-axis). You might be required to put the equation of a line into the slope-intercept form to determine either the slope or the y-intercept of a line.

For example: In the xy-plane, a line has the equation 3x + 7y – 16 = 0. What is the slope of the line?

The first step is to isolate y on the left side of the equation.

  • 3x + 7y – 16 = 0

  • 7y = -3x + 16

According to the slope-intercept formula, the slope of the line is .

Understanding Slope

The slope of a line is commonly defined as "rise over run," and is a value that is calculated by taking the change in y-coordinates divided by the change in x-coordinates from two given points on a line. The formula for slope is m = (y2y1) / (x2x1) where (x1, y1) and (x2, y2) are the two given points. For example, if you are given (3,2) and (5,6) as two points on a line, the slope would be m = (6 – 2) / (5 – 3) = = 2. A positive slope will mean that the

graph of the line will go up and to the right. A negative slope will mean that the graph of the line will go down and to the right. A horizontal line has a slope of 0, whereas a vertical line has an undefined slope (see Figure 3.2).

Figure 3.2

Figure 3.2 Slopes of a line.

Understanding Parallel and Perpendicular Lines

Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the slope of one of the lines is the negative reciprocal of the slope of the other line. If the slope of line a is 5, then the slope of line b must be - for the lines to be perpendicular (see Figure 3.3).

Figure 3.3

Figure 3.3 Parallel and perpendicular lines.

Understanding Distance and Midpoint Formulas

To find the distance between two points on a coordinate graph, use the formula , where (x1, y1) and (x2, y2) are the two given points. For instance, the distance between (3,2) and (7,6) is calculated as follows:

  • =

To find the midpoint of a line given two points on the line, use the formula . For example, the midpoint between (5,4) and (9,2) is , or (7,3).

Understanding Properties and Relations of Plane Figures

Plane figures include circles, triangles, rectangles, squares, and other parallelograms. This book assumes a basic understanding of the properties of plane figures.

Triangles

A triangle is a polygon with three sides and three angles. If the measure of all three angles in the triangle are the same and all three sides of the triangle are the same length, then the triangle is an equilateral triangle. If the measure of two of the angles and two of the sides of the triangle are the same, then the triangle is an isosceles triangle. A scalene triangle has three unequal sides.

The sum of the interior angles in a triangle is always 180°. If the measure of one of the angles in the triangle is 90° (a right angle), then the triangle is a right triangle, as shown in Figure 3.4.

Figure 3.4

Figure 3.4 Right triangle.

Some right triangles have unique relationships between the angles and the lengths of the sides. These are called "Special Right Triangles." It might be helpful to remember the information shown in Figure 3.5.

The perimeter of a triangle is the sum of the lengths of the sides. The area of a triangle is calculate by using the formula A = (base)(height). For any right triangle, the Pythagorean theorem states that a2 + b2 = c2, where a and b are legs (sides) and c is the hypotenuse.

Figure 3.5

Figure 3.5 Special right triangles.

Circles

The equation of a circle centered at the point (h,k) is (xh)2 + (yk)2 = r2, where r is the radius of the circle. The radius of a circle is the distance from the center of the circle to any point on the circle. The diameter of a circle is twice the radius. The formula for the circumference of a circle is C = 2πr, or πd, while the formula for the area of a circle is A = πr2. A circle contains 360°.

Rectangle

A rectangle, as shown in Figure 3.6, is a polygon with four sides (two sets of congruent, or equal sides) and four right angles. The sum of the angles in a rectangle is always 360°. The perimeter of a rectangle is P = 2l + 2w, where l is the length and w is the width. The area of a rectangle is A = lw. The lengths of the diagonals of a rectangle are congruent, or equal. A square is a special rectangle where all four sides are of equal length.

Figure 3.6

Figure 3.6 Rectangle.

Parallelograms

A parallelogram, shown in Figure 3.7, is a polygon with four sides and four angles, which are not right angles. A parallelogram has two sets of congruent sides and two sets of congruent angles. The sum of the angles of a parallelogram is 360°. The perimeter of a parallelogram is P = 2l + 2w. The area of a parallelogram is A = (base)(height). The height is the distance from top to bottom. A rhombus is a special parallelogram with four congruent sides.

Figure 3.7

Figure 3.7 Parallelogram.

Trapezoids

A trapezoid, as shown in Figure 3.8, is a polygon with four sides and four angles. The bases of the trapezoids (top and bottom) are never the same length. The sides of the trapezoid can be the same length (isosceles trapezoid), or they might not be. The perimeter of the trapezoid is the sum of the lengths of the sides. The area of a trapezoid is A = (base1 + base2)(height).

The height is the distance between the bases. The diagonals of a trapezoid have a unique feature. When the diagonals of a trapezoid intersect, the ratio of the top of the diagonals to the bottom of the diagonals is the same as the ratio of the top base to the bottom base.

Figure 3.8

Figure 3.8 Trapezoid.

Understanding Angles

Angles can be classified as acute, obtuse, or right. An acute angle is any angle less than 90°. An obtuse angle is any angle greater than 90° and less than 180°. A right angle is a 90° angle.

When two parallel lines are cut by a perpendicular line, right angles are created, as shown in Figure 3.9.

Figure 3.9

Figure 3.9 Parallel lines cut by a perpendicular line.

When two parallel lines are cut by a transversal, or intersecting line, the angles created have special properties. Each of the parallel lines cut by the transversal has four angles surrounding the intersection, that are matched in measure and position with a counterpart at the other parallel line. The vertical (opposite) angles are congruent, and the adjacent angles are supplementary; that is, the sum of the two supplementary angles is 180°. Figure 3.10 shows these special relationships.

Figure 3.10

Figure 3.10 Parallel lines cut by a transversal.

Understanding Perimeter, Area, and Volume

The area, perimeter, and volume of geometric figures involve the size and amount of space taken up by a particular figure. Several of these formulas are included in reference information that is printed in the directions on the PSAT math sections, but we will review the basic formulas here.

Perimeter

The formulas for calculating the perimeter of shapes that might appear on the PSAT math sections are as follows:

  1. Triangle: sum of the sides

  2. Rectangle and parallelogram: 2l + 2w

  3. Square: 4s (s is the length of each side)

  4. Trapezoid: sum of the sides

  5. Circle (Circumference): 2πr, or πd.

Area

The formulas for calculating the area of shapes that might appear on the PSAT math sections are as follows:

  1. Triangle: (base)(height)

  2. Rectangle and square: (length)(width)

  3. Parallelogram: (base)(height)

  4. Trapezoid: (base 1 + base 2)(height)

  5. Circle: πr2

Volume

The formulas for calculating the volume of basic three-dimensional shapes that might appear on the PSAT math sections are as follows:

  1. Rectangular box and cube: (length)(width)(height)

  2. Sphere:

  3. Right circular cylinder: πr2h (h is the height)

Understanding Word Problems

Many PSAT math questions are presented as word problems that require you to apply math skills to everyday situations. It is important that you carefully read the questions and understand what is being asked. Some of the information given might not be relevant to answering the question. To stay on track, you can cross out any information that is not necessary to solve the problem. The table shown in Figure 3.11 represents the relationship between some words and their mathematical counterparts.

Figure 3.11

Figure 3.11 Word problem translations.

Understanding Data Interpretation Problems

These questions require you to interpret information that is presented in charts, graphs, or tables, compare quantities, recognize trends and changes in the data, and perform basic calculations based on the information contained in the figures.

Pie Charts

Information can be presented in the form of a pie chart. Figure 3.12 is an example of a pie chart like one that you might see on the PSAT.

Figure 3.12

Figure 3.12  Sample pie chart.

Line Graphs

Line graphs can also be used to display information. Figure 3.13 shows an example of a line graph like one that you might see on the PSAT.

Figure 3.13

Figure 3.13 Sample line graph.

Bar Graphs

Data can also be represented on a bar graph. Figure 3.14 is an example of a bar graph like one that you might see on the PSAT.

Figure 3.14

Figure 3.14 Sample bar graph.

Pictographs

A pictograph displays statistical information using pictures or symbols. Figure 3.15 shows an example of a pictograph like one that you might see on the PSAT.

Figure 3.15

Figure 3.15 Sample pictograph.

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