Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-14: (a) Find the distance \(d(A, B)\) between \(A\) and \(B\). (b) Find the midpoint of the segment \(A B\). $$ A(7,-3), \quad B(3,-3) $$

Step-by-Step Solution

Answer

Distance: 4 units; Midpoint: (5, -3).

Step 1: Understand the Problem

We need to find two things: (a) the distance between the points \(A(7, -3)\) and \(B(3, -3)\), and (b) the midpoint of the segment connecting these two points.

Step 2: Write the Distance Formula

The distance \(d\) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \(d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Plug in the Coordinates for Distance

Substitute \(x_1 = 7\), \(y_1 = -3\), \(x_2 = 3\), and \(y_2 = -3\) into the distance formula.

Step 4: Calculate the Distance

Use the formula: \[ d(A, B) = \sqrt{(3 - 7)^2 + (-3 + 3)^2} = \sqrt{(-4)^2 + 0} = \sqrt{16 + 0} = \sqrt{16} = 4 \].Thus, the distance \(d(A, B)\) is 4 units.

Step 5: Write the Midpoint Formula

The midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).

Step 6: Plug in the Coordinates for Midpoint

Substitute \(x_1 = 7\), \(y_1 = -3\), \(x_2 = 3\), \(y_2 = -3\) into the midpoint formula.

Step 7: Calculate the Midpoint

Use the formula:\[ M = \left( \frac{7 + 3}{2}, \frac{-3 + (-3)}{2} \right) = \left( \frac{10}{2}, \frac{-6}{2} \right) = (5, -3) \].Thus, the midpoint is \((5, -3)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a critical mathematical tool used to study the position of points and the relationships between them on a plane. It involves the use of a coordinate system, usually the Cartesian plane, to locate points using ordered pairs of numbers. These numbers are called coordinates and are written as \( (x, y) \), where \( x \) represents the horizontal position, or the ‘x-coordinate,’ and \( y \) represents the vertical position, or the ‘y-coordinate.’

• Each point in this system corresponds to a unique pair of coordinates.
• The plane is divided into four quadrants by the x and y axes.

The foundation of coordinate geometry enables us to use simple algebraic formulas to determine distances, midpoints, and other geometric properties. This method simplifies the process of understanding and solving geometrical problems by converting them into algebraic expressions.

Distance Formula

The Distance Formula is a mathematical equation used to calculate the distance between two points on the Cartesian plane. It is derived using the Pythagorean theorem, which is an important principle in geometry for finding the length of a side of a right triangle. The formula helps us easily determine how far apart two points are.
The formula is written as:
\[d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Here's how you can apply the Distance Formula:

• Identify the coordinates of the two points. For example, if you have two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), substitute these values into the formula.
• Solve the equation by finding the square of the differences in the \( x \) and \( y \) coordinates, then add these values together.
• Take the square root of the sum to find the distance between the points.

In the example given, with points \( A(7, -3) \) and \( B(3, -3) \), the distance calculated is 4 units.

Midpoint Formula

The Midpoint Formula is used to find the exact center between two points on a Cartesian plane. It gives us the average of the x and y coordinates of these points, resulting in a new point that lies exactly halfway between the initial points. This formula is particularly useful in finding the "middle" location along a line segment.
The formula is expressed as:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

• To use the formula, take the x-coordinates of both points, add them together, and divide by 2. This results in the x-coordinate of the midpoint.
• Do the same for the y-coordinates to find the y-coordinate of the midpoint.

For instance, with endpoints \( A(7, -3) \) and \( B(3, -3) \), the midpoint found is \((5, -3)\), which indicates that the midpoint lies directly between the two points on the plane.