Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

Group icon
40,131 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on developing the ability to calculate distances and midpoints in the coordinate plane using the Distance and Midpoint Formulas. Derived from the principles of the Pythagorean Theorem, these formulas provide essential tools for bridging algebra and geometry. Mastery of these concepts not only aids in solving standard geometric problems, such as verifying right triangles and computing areas, but also has practical applications in various real-world contexts.

Learning Objectives

1

Use the Distance Formula to compute the distance between two points in the coordinate plane.

2

Apply the Midpoint Formula to determine the center point of a line segment.

3

Utilize the Distance and Midpoint Formulas to verify geometric properties, such as identifying right triangles and calculating areas.

4

Translate between algebraic and geometric representations by plotting points and forming right triangles for problem solving.

Key Concepts

CONCEPT

DEFINITION

Distance Formula

A formula used to calculate the distance between two points (x1, y1) and (x2, y2) in the Cartesian plane, given by d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Midpoint Formula

A formula that gives the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2), calculated as M = ((x1 + x2)/2, (y1 + y2)/2).

Pythagorean Theorem

A principle in geometry stating that in a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2, which underpins the derivation of the Distance Formula.

Right Triangle

A triangle that has one 90° angle, often verified by checking if the squares of two sides sum to the square of the third side.

Viewing Rectangle

The portion of the coordinate plane displayed on a graphing calculator, defined by the scales set on the x-axis and y-axis.

Example Problems

Example 1

On the real number line the origin is assigned the number______.

Example 2

If $-3$ and 5 are the coordinates of two points on the real number line, the distance between these points is___.

Example 3

If 3 and 4 are the legs of a right triangle, the hypotenuse is___.

Example 4

Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths $11,60,$ and 61 is a right triangle.

Example 5

The area $A$ of a triangle whose base is $b$ and whose altitude is $h$ is $A=$ _____.
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Find the distance between the points (15, 62) and (11, 32).

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: (x1, y1) = (15, 62) and (x2, y2) = (11, 32).
Step 2: Compute the difference in the x-coordinates: 11 - 15 = -4. Then square it: (-4)^2 = 16.
Step 3: Compute the difference in the y-coordinates: 32 - 62 = -30. Then square it: (-30)^2 = 900.
Step 4: Add these squared differences: 16 + 900 = 916.
Step 5: Take the square root of the sum: d = sqrt(916).
Final Answer: The distance between the points is sqrt(916), which can be simplified if needed.

Distance Formula

QUESTION

Find the midpoint of the line segment joining the points (-5, 5) and (3, 1).

STEP-BY-STEP ANSWER:

Step 1: Identify the endpoints: (x1, y1) = (-5, 5) and (x2, y2) = (3, 1).
Step 2: Compute the average of the x-coordinates: (-5 + 3) / 2 = (-2) / 2 = -1.
Step 3: Compute the average of the y-coordinates: (5 + 1) / 2 = 6 / 2 = 3.
Final Answer: The midpoint is (-1, 3).

Midpoint Formula

Scroll left
Scroll right

Common Mistakes

  • Forgetting to square the differences before adding them when using the Distance Formula.
  • Mixing up the order of subtraction, particularly with negative coordinate values.
  • Incorrectly averaging the coordinates when computing the midpoint, such as dividing by the wrong number.
  • Assuming that the order of points changes the result, although distance is independent of order.
  • Overlooking the importance of plotting points accurately, which can lead to misinterpretation of geometric figures.