Problem 90 - A Graphical Approach to College Algebra

Question

Find (a) the distance between P and Q and (b) the coordinates of the midpoint M of the segment joining P and Q $$P(-2,5), Q(4,-3)$$

Step-by-Step Solution

Answer

Distance = 10 units, Midpoint = (1, 1).

Step 1: Use the Distance Formula

To find the distance between points \( P(-2,5) \) and \( Q(4,-3) \), use the distance formula. The distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute \(x_1 = -2, y_1 = 5, x_2 = 4,\) and \(y_2 = -3\):\[ d = \sqrt{(4 - (-2))^2 + (-3 - 5)^2} \]\[ = \sqrt{(4 + 2)^2 + (-8)^2} \]\[ = \sqrt{6^2 + 8^2} \]\[ = \sqrt{36 + 64} \]\[ = \sqrt{100} \]\[ = 10 \]The distance between \( P \) and \( Q \) is 10 units.

Step 2: Use the Midpoint Formula

To find the midpoint \( M \) of the segment joining \( P(-2,5) \) and \( Q(4,-3) \), use the midpoint formula:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Substitute \(x_1 = -2, y_1 = 5, x_2 = 4,\) and \(y_2 = -3\):\[ M = \left( \frac{-2 + 4}{2}, \frac{5 + (-3)}{2} \right) \]\[ = \left( \frac{2}{2}, \frac{2}{2} \right) \]\[ = (1, 1) \]The coordinates of the midpoint \( M \) are \((1, 1)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of geometry where the position of points on a plane is described using ordered pairs called coordinates. The coordinate plane is like a map where every point has a specific address, found by moving along two perpendicular (right-angled) lines called axes. These axes are known as the x-axis (horizontal) and the y-axis (vertical). At the intersection of these two axes is the origin, with coordinates (0, 0). This system allows for the graphical representation of geometric figures and the solving of various mathematical problems.

Points on this plane are expressed as pairs

• The first number represents the position along the x-axis
• The second number represents the position along the y-axis

This idea is central to calculating distances and midpoints between two points, as you can see in the exercise. When you have coordinates for any two points, you can determine the spatial relationship between these points through mathematical formulas.

Distance Formula

The Distance Formula allows us to calculate how far apart two points are on the coordinate plane. Imagine you are looking for the shortest path between two locations on a map. The distance formula is an algebraic method to find that shortest path. The formula comes from the Pythagorean Theorem, which relates to right triangles.

To use the formula, you need the coordinates of both points:

• For a point \( P(x_1, y_1) \)
• And a point \( Q(x_2, y_2) \)

The distance \( d \) is calculated as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This formula calculates the length of the hypotenuse of a right triangle, which is the distance between the two points. It considers the difference in the x-coordinates (\( x_2 - x_1 \)) and the difference in the y-coordinates (\( y_2 - y_1 \)), forming two sides of the triangle. The hypotenuse, found using the square root, is the direct distance between the points.

Midpoint Formula

The Midpoint Formula helps us find the exact middle point of a line segment connecting two points on a coordinate plane. Think of it as finding the center point, or average point, along a line. The midpoint is important in various contexts, such as dividing land in equal halves or constructing symmetrical shapes.

To calculate the midpoint, you need the coordinates of the two endpoints, \( P(x_1, y_1) \) and \( Q(x_2, y_2) \). The formula for the midpoint \( M \) is:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This formula takes the average of the x-coordinates and y-coordinates separately. It gives you a new point that is exactly halfway between the original two points. Consider it as balancing the weights of both coordinates equally, resulting in a point right in the center.