Problem 93 - 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 $$P(9.2,3.4), Q(6.2,7.4)$$

Step-by-Step Solution

Answer

The distance between P and Q is 5 units, and the midpoint M is (7.7, 5.4).

Step 1: Understand the Distance Formula

The distance between two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) in a plane is given by the formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]This formula will help us find the distance between points P(9.2, 3.4) and Q(6.2, 7.4).

Step 2: Calculate the Distance

Substitute the coordinates of P(9.2, 3.4) and Q(6.2, 7.4) into the distance formula: \[d = \sqrt{(6.2 - 9.2)^2 + (7.4 - 3.4)^2}\]Now, calculate:\[d = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]Thus, the distance between P and Q is 5 units.

Step 3: Understand the Midpoint Formula

The midpoint M of a segment connecting two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by the formula: \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]. This formula provides the average of the x-coordinates and the y-coordinates of the endpoints to find the coordinates of the midpoint.

Step 4: Calculate the Midpoint

Use the midpoint formula with the coordinates of P(9.2, 3.4) and Q(6.2, 7.4): \[M = \left(\frac{9.2 + 6.2}{2}, \frac{3.4 + 7.4}{2}\right)\]Calculate each coordinate:\[x-coordinate: \frac{9.2 + 6.2}{2} = \frac{15.4}{2} = 7.7\]\\[y-coordinate: \frac{3.4 + 7.4}{2} = \frac{10.8}{2} = 5.4\]Thus, the coordinates of the midpoint M are (7.7, 5.4).

Key Concepts

Distance Formula

Calculating the distance between two points in a plane is a common problem in coordinate geometry, and the Distance Formula is an essential tool for solving it. The formula is derived from the Pythagorean theorem, which works in the context of a right triangle. This makes sense because in coordinate geometry, a pair of points can form the legs of a right triangle, with the distance between them as the hypotenuse.

The formula for the distance between two points, \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), is:

• \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

Here’s how you can use it:

• Subtract the x-coordinates: \(x_2 - x_1\)
• Subtract the y-coordinates: \(y_2 - y_1\)
• Square both differences to eliminate negative numbers
• Add the squared differences
• Take the square root to find the distance

By applying this process to points \(P(9.2, 3.4)\) and \(Q(6.2, 7.4)\), you calculated the distance to be 5 units. Notice how the steps naturally guide you through breaking down the process into manageable parts.

Midpoint Formula

The midpoint of a line segment is a point that divides the segment into two equal halves. To find this point precisely, we use the Midpoint Formula. This formula provides a way to find the average position of the endpoints of the segment in a coordinate system.

The Midpoint Formula for two points, \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), is:

• \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

Here's a simple breakdown:

• Add the x-coordinates together and divide by 2, giving you the x-coordinate of the midpoint.
• Add the y-coordinates together and divide by 2, giving you the y-coordinate of the midpoint.

For points \(P(9.2, 3.4)\) and \(Q(6.2, 7.4)\), using the midpoint formula gives the coordinates of the midpoint as \((7.7, 5.4)\). By averaging the coordinates, you effectively find the center point of the line segment.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, uses the principles of geometry and algebra together to describe and analyze geometric shapes within a coordinate plane. Understanding this area of mathematics provides a deeper insight into the spatial relationships of different geometric figures.

Key concepts in coordinate geometry include:

• Points: Defined by coordinates \((x, y)\) in the plane.
• Distance: Calculated between points using the distance formula. This helps in quantifying how far apart two points are.
• Midpoint: Found using the midpoint formula, providing a "balancing point" of a segment.
• Slopes and Lines: Equations of lines such as slope-intercept form \(y = mx + b\) help describe the orientation and position of a line in space.

Coordinate geometry allows us to plot and manipulate figures using their algebraic representations, thereby solving problems related to distances, midpoints, lines, and shapes on the plane easily and effectively.