Unit 1 Geometry Basics Homework 3 Distance And Midpoint Formulas

Unit 1 Geometry Basics Homework 3: Mastering Distance and Midpoint Formulas

This comprehensive guide delves into the intricacies of the distance and midpoint formulas, crucial components of Unit 1: Geometry Basics. We'll explore these formulas in detail, providing numerous examples and practice problems to solidify your understanding. By the end, you'll be confidently tackling any distance and midpoint formula problem thrown your way. Let's begin!

Understanding the Coordinate Plane

Before diving into the formulas, a strong grasp of the coordinate plane is essential. The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Their intersection point is called the origin (0,0). Each point on the plane is represented by an ordered pair (x, y), where 'x' represents the horizontal distance from the origin and 'y' represents the vertical distance from the origin.

Key Coordinate Plane Terminology:

• Quadrants: The coordinate plane is divided into four quadrants, numbered I, II, III, and IV, counterclockwise starting from the top right.
• Ordered Pairs: Points are represented as ordered pairs (x, y). The order matters! (2, 3) is different from (3, 2).
• Origin: The point (0, 0), where the x-axis and y-axis intersect.

The Distance Formula: Calculating the Distance Between Two Points

The distance formula is used to find the distance between any two points on the coordinate plane. It's derived from the Pythagorean theorem, a fundamental concept in geometry.

The Formula:

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Breaking Down the Formula:

1. (x₂ - x₁)²: Find the difference between the x-coordinates, square the result.
2. (y₂ - y₁)²: Find the difference between the y-coordinates, square the result.
3. (x₂ - x₁)² + (y₂ - y₁)²: Add the squared differences.
4. √[(x₂ - x₁)² + (y₂ - y₁)²]: Take the square root of the sum to find the distance.

Example 1: Finding the Distance Between Two Points

Let's find the distance between points A(1, 2) and B(4, 6).

1. x₂ - x₁ = 4 - 1 = 3
2. y₂ - y₁ = 6 - 2 = 4
3. (x₂ - x₁)² = 3² = 9
4. (y₂ - y₁)² = 4² = 16
5. (x₂ - x₁)² + (y₂ - y₁)² = 9 + 16 = 25
6. d = √25 = 5

Therefore, the distance between points A and B is 5 units.

Example 2: A More Complex Example

Let's find the distance between points C(-2, 3) and D(5, -1).

1. x₂ - x₁ = 5 - (-2) = 7
2. y₂ - y₁ = -1 - 3 = -4
3. (x₂ - x₁)² = 7² = 49
4. (y₂ - y₁)² = (-4)² = 16
5. (x₂ - x₁)² + (y₂ - y₁)² = 49 + 16 = 65
6. d = √65

The distance between points C and D is √65 units. Note that you can leave the answer in radical form unless specifically instructed otherwise.

The Midpoint Formula: Finding the Middle Point

The midpoint formula helps us locate the exact middle point between two points on the coordinate plane.

The Formula:

The midpoint M between two points (x₁, y₁) and (x₂, y₂) is given by:

M = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Breaking Down the Formula:

1. (x₁ + x₂)/2: Find the average of the x-coordinates.
2. (y₁ + y₂)/2: Find the average of the y-coordinates.
3. [(x₁ + x₂)/2, (y₁ + y₂)/2]: The resulting ordered pair represents the midpoint.

Example 3: Finding the Midpoint Between Two Points

Let's find the midpoint between points E(2, 4) and F(8, 10).

1. (x₁ + x₂)/2 = (2 + 8)/2 = 10/2 = 5
2. (y₁ + y₂)/2 = (4 + 10)/2 = 14/2 = 7

Therefore, the midpoint M between E and F is (5, 7).

Example 4: Midpoint with Negative Coordinates

Let's find the midpoint between points G(-3, 5) and H(1, -1).

1. (x₁ + x₂)/2 = (-3 + 1)/2 = -2/2 = -1
2. (y₁ + y₂)/2 = (5 + (-1))/2 = 4/2 = 2

Therefore, the midpoint M between G and H is (-1, 2).

Combining Distance and Midpoint Formulas: Advanced Applications

Many geometry problems require the use of both formulas in conjunction. Let's explore such scenarios.

Example 5: Finding the Length of a Segment Given its Midpoint

Let's say we know the coordinates of one endpoint of a line segment, A(1, 3), and its midpoint M(4, 6). We need to find the coordinates of the other endpoint, B(x, y), and the length of the segment AB.

1. Using the Midpoint Formula: We can set up equations using the midpoint formula:

   • (1 + x)/2 = 4 => 1 + x = 8 => x = 7
   • (3 + y)/2 = 6 => 3 + y = 12 => y = 9 Therefore, the coordinates of point B are (7, 9).

2. Using the Distance Formula: Now, we can use the distance formula to find the length of AB:

   • d = √[(7 - 1)² + (9 - 3)²] = √[6² + 6²] = √72 = 6√2

Therefore, the length of segment AB is 6√2 units.

Example 6: Verifying Properties of Geometric Figures

The distance and midpoint formulas are instrumental in verifying properties of geometric shapes. For instance, you could use them to prove that a quadrilateral is a parallelogram (opposite sides are equal and parallel), a rectangle (opposite sides are equal and all angles are 90 degrees), or a rhombus (all sides are equal). These proofs often involve calculating distances between vertices and finding midpoints of diagonals.

Practice Problems: Test Your Knowledge

Now it's time to test your skills! Try solving these problems:

1. Find the distance between the points (2, -5) and (7, 3).
2. Find the midpoint of the line segment connecting the points (-4, 6) and (10, -2).
3. One endpoint of a line segment is (5, -1) and its midpoint is (3, 2). Find the coordinates of the other endpoint.
4. Determine whether the triangle with vertices A(1, 2), B(4, 6), and C(7, 4) is an isosceles triangle (two sides of equal length).
5. Prove that the diagonals of a rectangle bisect each other (cut each other in half).

Conclusion: Mastering Geometry Basics

The distance and midpoint formulas are fundamental tools in coordinate geometry. By understanding their derivations and applications, you'll be well-equipped to solve a wide range of problems, from simple distance calculations to more complex geometric proofs. Consistent practice is key to mastering these formulas and building a solid foundation in geometry. Remember to review the examples and attempt the practice problems to solidify your understanding. Good luck!