Geometry for Nursing School Entrance Exam Study Guide (page 5)
The practice quiz for this study guide can be found at:
Geometry questions cover points, lines, planes, angles, triangles, rectangles, squares, and circles. You may be asked to determine the area or perimeter of a particular shape, the size of an angle, the length of a line, and so forth. Some word problems may also involve geometry.
Points, Lines, and Planes
What Is a Point?
A point has position but no size or dimension. It is usually represented by a dot named with an uppercase letter:
What Is a Line?
A line consists of an infinite number of points that extend endlessly in both directions.
A line can be named in two ways:
- By a letter at one end (typically in lowercase): l
- By two points on the line:
The following terminology is frequently used on math tests:
- Points are collinear if they lie on the same line. Points J, U, D, and I are collinear.
- A line segment is a section of a line with two endpoints. The line segment at right is indicated as .
- The midpoint is a point on a line segment that divides it into two line segments of equal length. M is the midpoint of line segment AB. Two line segments of the same length are said to be congruent. Congruent line segments are indicated by the same mark on each line segment (like the double marks shown below on ).
- A line segment (or line) that divides another line segment into two congruent line segments is said to bisect it. bisects .
- A ray is a section of a line that has one endpoint. The ray at the right is indicated as .
What Is a Plane?
A plane is like a flat surface with no thickness. Although a plane extends endlessly in all directions, it is usually represented by a four-sided figure and named by an uppercase letter in a corner of the plane: K.
Points are coplanar if they lie on the same plane. Points A and B are coplanar.
An angle is formed when two lines, segments, or rays meet at a point: The lines are called the sides of the angle, and the point where they meet is called the vertex of the angle.
The symbol used to indicate an angle is .
There are three ways to name an angle:
- By the letter that labels the vertex: B
- By the three letters that label the angle: ABC or CBA, with the vertex letter in the middle
- By the number inside the vertex: 1
An angle's size is based on the opening between its sides. Size is measured in degrees (°). The smaller the angle, the fewer degrees it has. Angles are classified by size. Notice how the arc () shows which of the two angles is indicated:
Special Angle Pairs
- Congruent angles: Two angles that have the same degree measure.
- Complementary angles: Two angles whose sum is 90°.
- Supplementary angles: Two angles whose sum is 180°.
- Vertical angles: Two angles that are opposite each other when two lines cross. Two sets of vertical angles are formed:
- 1 and 4
- 2 and 3
Congruent angles are indicated by identical markings.
The symbol is used to indicate that two angles are congruent: A B.
ABD and DBC are complementary angles.
ABD is the complement of DBC, and vice versa.
ABD and DBC are supplementary angles.
ABD is the supplement of DBC, and vice versa.
Hint: To avoid confusion between complementary and supplementary: C comes before S in the alphabet, and 90 comes before 180.
- Complementary: 90°
- Supplementary: 180°
Vertical angles are congruent.
When two lines cross, the adjacent angles are supplementary and the sum of all four angles is 360°.
Angle-pair problems tend to ask for an angle's complement or supplement.
Example: If the measure of 2 = 70°, what are the measures of the other three angles?
- 2 3 because they're vertical angles.
- Therefore, 3 = 70°.
- 1 and 2 are adjacent angles and therefore supplementary.
- Thus, 1 = 110° (180° – 70° = 110°).
- 1 4 because they're also vertical angles.
- Therefore, 4 = 110°.
Check: Add the angles to be sure their sum is 360°.
To solve geometry problems more easily, draw a picture if one is not provided. Try to draw the picture to scale. As the problem presents information about the size of an angle or line segment, label the corresponding part of your picture to reflect the given information. As you begin to find the missing information, label your picture accordingly.
Special Line Pairs
Parallel lines lie in the same plane and never cross at any point.
The arrowheads on the lines indicate that they are parallel. The symbol || is used to indicate that two lines are parallel: l || m.
When two parallel lines are crossed by another line, two groups of four angles each are formed. One group consists of 1, 2, 3, and 4; the other group contains 5, 6, 7, and 8.
These angles have special relationships:
- The four obtuse angles are always congruent: 1 4 5 8.
- The four acute angles are always congruent: 2 3 6 7.
- The sum of any one acute angle and any one obtuse angle is always 180° because the acute angles lie on the same line as the obtuse angles.
Don't be fooled into thinking two lines are parallel just because they look parallel. Either the lines must be marked with similar arrowheads or there must be an angle pair as just described.
Perpendicular lines lie in the same plane and cross to form four right angles.
The little box where the lines cross indicates a right angle. Because vertical angles are equal and the sum of all four angles is 360°, each of the four angles is a right angle and 90°. However, only one little box is needed to indicate this.
The symbol is used to indicate that two lines are perpendicular: .
Don't be fooled into thinking two lines are perpendicular just because they look perpendicular. The problem must indicate the presence of a right angle (by stating that an angle measures 90° or by the little right angle box in a corresponding diagram), or you must be able to prove the presence of a 90° angle.
A polygon is a closed, plane (flat) figure formed by three or more connected line segments that don't cross each other. Familiarize yourself with the following polygons; they are the four most common polygons appearing on standardized tests—and in life.
Perimeter is the distance around a polygon. The word perimeter is derived from peri, which means around (as in periscope and peripheral vision), and meter, which means measure. Thus perimeter is the measure around something. There are many everyday applications of perimeter. For instance, a carpenter measures the perimeter of a room to determine how many feet of ceiling molding she needs. A farmer measures the perimeter of a field to determine how many feet of fencing he needs to surround it.
Perimeter is measured in length units, like feet, yards, inches, meters, etc.
Example: Find the perimeter of the following polygon:
Write down the length of each side and add:
Note: The notion of perimeter also applies to a circle; however, the perimeter of a circle is referred to as its circumference. We will take a closer look at circles and circumference later in this chapter.
Area is the total amount of space taken by a figure's surface. Area is measured in square units. For instance, a square that is 1 unit on all sides covers 1 square unit. If the unit of measurement for each side is feet, for example, then the area is measured in square feet; other possibilities are units like square inches, square miles, square meters, and so on.
You could measure the area of any figure by counting the number of square units the figure occupies. The first two figures are easy to measure because the square units fit into them evenly, while the following two figures are more difficult to measure because the square units don't fit into them evenly.
Because it's not always practical to measure a particular figure's area by counting the number of square units it occupies, an area formula is used. As each figure is discussed, you'll learn its area formula. Although there are perimeter formulas as well, you don't really need them (except for circles) if you understand the perimeter concept: It is merely the sum of the lengths of the sides.
A triangle is a polygon with three sides, like those shown here:
The symbol used to indicate a triangle is Δ. Each vertex—the point at which two lines meet—is named by a capital letter. The triangle is named by the three letters at the vertices, usually in alphabetical order: ΔABC.
There are two ways to refer to a side of a triangle:
- By the letters at each end of the side: AB
- By the letter—typically a lowercase letter—next to the side: c
Notice that the name of the side is the same as the name of the angle opposite it, except the angle's name is a capital letter.
There are two ways to refer to an angle of a triangle:
- By the letter at the vertex: A
- By the triangle's three letters, with that angle's vertex letter in the middle: BAC or CAB
Types of Triangles
A triangle can be classified by the size of its angles and sides.
- three congruent angles, each 60°
- three congruent sides
Hint to help you remember: The word equilateral comes from equi, meaning equal, and lat, meaning side. Thus, all equal sides.
- two congruent angles, called base angles; the third angle is the vertex angle.
- Sides opposite the base angles are also congruent.
- An equilateral triangle is also isosceles, since it always has two congruent angles.
- one right (90°) angle, the largest angle in the triangle
- The side opposite the right angle is the hypotenuse, the longest side of the triangle. (Hint: The word hypotenuse reminds us of hippopotamus, a very large animal.)
- The other two sides are called legs.
Area of a Triangle
To find the area of a triangle, use this formula:
Although any side of a triangle may be called its base, it's often easiest to use the side on the bottom. To use another side, rotate the page and view the triangle from another perspective.
A triangle's height is represented by a perpendicular line drawn from the angle opposite the base to the base. Depending on the triangle, the height may be inside, outside, or on the triangle. Notice the height of the second triangle: We extended the base to draw the height perpendicular to the base. The third triangle is a right triangle: One leg may be its base and the other its height.
Hint: Think of a triangle as being half a rectangle. The area of that triangle is half the area of the rectangle.
Example: Find the area of a triangle with a 2-inch base and a 3-inch height.
- Draw the triangle as close to scale as you can.
- Label the size of the base and height.
- Write the area formula; then substitute the base and height numbers into it:
- The area of the triangle is 3 square inches.
The following rules tend to appear more frequently on standardized tests than other rules. A typical test question follows each rule.
Example: One base angle of an isosceles triangle is 30°. Find the vertex angle.
- Draw a picture of an isosceles triangle. Drawing it to scale helps: Since it is an isosceles triangle, draw both base angles the same size (as close to 30° as you can) and make sure the sides opposite them are the same length. Label one base angle as 30°.
- Since the base angles are congruent, label the other base angle as 30°.
- There are two steps needed to find the vertex angle:
- Add the two base angles together: 30° + 30° = 60°
- The sum of all three angles is 180°. To find the vertex angle, subtract the sum of the two base angles (60°) from 180°: 180° – 60° = 120°
Thus, the vertex angle is 120°.
Check: Add all three angles together to make sure their sum is 180°:
- 30° + 30° + 120° = 180°
Example: In the triangle shown at the right, which side is the shortest?
- Determine the size of A, the missing angle, by adding the two known angles and then subtracting their sum from 180°: Thus, A is 44°.
- Since A is the smallest angle, side BC, which is opposite A, is the shortest side.
Example: What is the perimeter of the triangle shown at the right?
- Since the perimeter is the sum of the lengths of the sides, we must first find the missing side. Use the Pythagorean theorem since you know this is a right triangle.
- Substitute the given sides for two of the letters. Remember: Side c is always the hypotenuse.
- To solve this equation, subtract 9 from both sides:
- Then, take the square root of both sides. Thus, the missing side has a length of 4 units:
- Adding the three sides yields a perimeter of 12:
A radical is simplified if there is no perfect square factor of the radicand. For example, is simplified because 10 has no perfect square factors. But, is not simplified because 20 has a perfect square factor of 4.
In order to simplify a radical, rewrite the radical as the product of two radicals, one of which is the largest perfect square factor of the radicand. The square root of a perfect square always simplifies to a rational number. Simplify the perfect square radical to get your final answer.
Example: Simplify .
A quadrilateral is a four-sided polygon. Following are examples of quadrilaterals that are most likely to appear on standardized tests (and in everyday life):
These quadrilaterals have something in common beside having four sides:
- Opposite sides are the same size and parallel.
- Opposite angles are the same size.
However, each quadrilateral has its own distinguishing characteristics:
The naming conventions for quadrilaterals are similar to those for triangles:
- The figure is named by the letters at its four consecutive corners, usually in alphabetic order: rectangle ABCD.
- A side is named by the letters at its ends: side AB.
- An angle is named by its vertex letter: A.
The sum of the angles of a quadrilateral is 360°: A + B + C + D = 360°
To find the perimeter of a quadrilateral, follow this simple rule:
Shortcut: Take advantage of the fact that the opposite sides of a rectangle and a parallelogram are equal: Just add two adjacent sides and double the sum. Similarly, multiply one side of a square by four.
To find the area of a rectangle, square, or parallelogram, use this formula:
The base is the size of one of the sides. It is easiest if you call the side on the bottom the base, but any side can be a base. The height (or altitude) is the size of a perpendicular line drawn from the base to the side opposite it. The height of a rectangle and a square is the same as the size of its non-base side.
Caution: A parallelogram's height is not usually the same as the size connecting the base to its opposite side (called the slant height), but the size of a perpendicular line drawn from the base to the side opposite it.
Example: Find the area of a rectangle with a base of 4 meters and a height of 3 meters.
- Draw the rectangle as close to scale as possible.
- Label the size of the base and height.
- Write the area formula; then substitute the base and height numbers into it: Thus, the area is 12 square meters.
We can all recognize a circle when we see one, but its definition is a bit technical. A circle is a set of points that are all the same distance from a given point called the center. That distance is called the radius. The diameter is twice the length of the radius; it passes through the center of the circle.
The circumference of a circle is the distance around the circle (it is the perimeter of the circle). To determine the circumference of a circle, use either of these two equivalent formulas:
- r is the radius
- d is the diameter (which is the same as 2x the radius)
- π is approximately equal (denoted by the symbol ) to 3.14 or
Note: Math often uses letters of the Greek alphabet, like π (pi). Perhaps that's what makes math seem like Greek to some people! In the case of the circle, you can use π as a hint to recognize a circle question: A pie is shaped like a circle.
Example: Find the circumference of a circle whose radius is 7 inches.
- Draw this circle and write the radius version of the circumference formula (because you're given the radius):
- Substitute 7 for the radius:
- On a multiple-choice test, look at the answer choices to determine whether to leave π in your answer or substitute the value of π in the formula. If the answer choices don't include π, substitute or 3.14 for π and multiply:
- If the answer choices include π, just multiply:
All the answers—44 inches, 43.96 inches, and 14π inches—are considered correct.
Example: What is the diameter of a circle with a circumference of 62.8 centimeters? Use 3.14 for π.
- Draw a circle with its diameter and write the diameter version of the circumference formula (because you're asked to find the diameter):
- Substitute 62.8 for the circumference, 3.14 for π, and solve the equation:
- 62.8 = 3.14 × d
- 62.8 = 3.14 × 20
The diameter is 20 centimeters.
The area of a circle is the space its surface occupies. To determine the area of a circle, use this formula:
Hint: To avoid confusing the area and circumference formulas, just remember that area is always measured in square units, like 12 square yards of carpeting. Thus, the area formula is the one with the squared term in it.
Example: Find the area of the circle at right, rounded to the nearest tenth:
- Write the area formula:
- Substitute 2.3 for the radius:
- On a multiple-choice test, look at the answer choices to determine whether to use π or an approximate value of π (decimal or fraction) in the formula. If the answers don't include π, use 3.14 for π (because the radius is a decimal):
If the answers include π, multiply and round:
Both answers—16.6 square inches and 5.3π square inches—are correct.
Example: What is the diameter of a circle with an area of 9π square centimeters?
- Draw a circle with its diameter (to help you remember that the question asks for the diameter); then write the area formula:
- Substitute 9π for the area and solve the equation:
Since the radius is 3 centimeters, the diameter is 6 centimeters.
The practice quiz for this study guide can be found at:
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