Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the Distance Formula and Midpoint Formula, fundamental tools in coordinate geometry. The Distance Formula, derived from the Pythagorean Theorem, allows us to calculate the straight-line distance between any two points in the plane. The Midpoint Formula provides a method to locate the exact center of a line segment. Both methods are not only essential for solving geometric problems but also have a range of practical applications in various real-world contexts.

Learning Objectives

1

Apply the Distance Formula to calculate the distance between any two points in the coordinate plane.

2

Use the Midpoint Formula to determine the midpoint of a line segment given its endpoints.

3

Derive and explain the Distance Formula using the Pythagorean Theorem.

4

Analyze geometric problems by translating them into algebraic solutions using these formulas.

5

Verify properties of triangles (such as right triangles) using the relationships derived from the distance calculations.

Key Concepts

CONCEPT

DEFINITION

Distance Formula

A formula to calculate the distance between two points (x1, y1) and (x2, y2) in the coordinate plane: d = √[(x2 - x1)² + (y2 - y1)²].

Midpoint Formula

A formula to find the midpoint of a line segment with endpoints (x1, y1) and (x2, y2): M = ((x1 + x2)/2, (y1 + y2)/2).

Pythagorean Theorem

A theorem stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Right Triangle

A triangle in which one of the angles is exactly 90 degrees. It satisfies the condition that the sum of the squares of the lengths of the two legs equals the square of the hypotenuse.

Coordinates

An ordered pair (x, y) that represents a point’s location in the Cartesian plane using a horizontal (x-axis) and vertical (y-axis) measurement.

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 $A=$ _____

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Step-by-Step Explanations

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates. Let (x1, y1) = (11, 32) and (x2, y2) = (15, 62).
Step 2: Compute the differences: x2 - x1 = 15 - 11 = 4, and y2 - y1 = 62 - 32 = 30.
Step 3: Square the differences: 4² = 16 and 30² = 900.
Step 4: Sum the squares: 16 + 900 = 916.
Step 5: Take the square root: d = √916. If necessary, simplify or approximate √916 ≈ 30.27.
Final Answer: The distance between the points is approximately 30.27 units.

Distance between two points

QUESTION

Find the midpoint M of the line segment joining P1 = (-5, 5) and P2 = (3, 1).

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: (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 M is (-1, 3).

Midpoint of a line segment

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Common Mistakes

  • Mixing up the order of subtraction: The formula uses (x2 - x1) and (y2 - y1), but the values are squared so order won’t affect the final distance – however, consistently keeping track of coordinates is important for clarity.
  • Forgetting to square the differences before taking their sum.
  • Neglecting to take the square root after adding the squares when using the Distance Formula.
  • Incorrectly averaging coordinates, such as adding coordinates without dividing by 2, when finding the midpoint.
  • Misidentifying the coordinate axes when plotting points, which can lead to errors in plotting the correct position or quadrant.