Problem 96 - 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. $$P(12 y,-3 y), Q(20 y, 12 y), \quad y>0$$

Step-by-Step Solution

Answer

(a) Distance = \( 17y \). (b) Midpoint \( M = \left(16y, \frac{9y}{2}\right) \).

Step 1: Find the distance

\( d = \sqrt{(20y-12y)^2 + (12y-(-3y))^2} = \sqrt{(8y)^2 + (15y)^2} = \sqrt{64y^2 + 225y^2} = \sqrt{289y^2} = 17y \) (since \( y > 0 \)).

Step 2: Find the midpoint

\( M = \left(\frac{12y+20y}{2}, \frac{-3y+12y}{2}\right) = \left(\frac{32y}{2}, \frac{9y}{2}\right) = \left(16y, \frac{9y}{2}\right) \).

Key Concepts

Distance Formula

The Distance Formula is a key tool in coordinate geometry, used to calculate the distance between two points on a coordinate plane. In a Cartesian coordinate system, each point is identified by an ordered pair of numbers (x, y). To find the distance between two points, like P(12y, -3y) and Q(20y, 12y), you can employ the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here:

• \(x_1\), \(y_1\) are the coordinates of the first point (P).
• \(x_2\), \(y_2\) are the coordinates of the second point (Q).

Substituting the given coordinates into the formula:

• \(d = \sqrt{(20y - 12y)^2 + (12y - (-3y))^2} = \sqrt{(8y)^2 + (15y)^2}\)
• \(d = \sqrt{64y^2 + 225y^2} = \sqrt{289y^2} = 17y\)

This formula is an extension of the Pythagorean theorem, providing a way to connect geometry with algebra.

Midpoint Formula

The Midpoint Formula helps you find the center point of a line segment connecting two points in a coordinate plane. It essentially averages the x-coordinates and y-coordinates of the two endpoints. The midpoint is represented as:\[M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Given points P(12y, -3y) and Q(20y, 12y):

• The average of the x-coordinates is \(\frac{12y + 20y}{2} = \frac{32y}{2} = 16y\)
• The average of the y-coordinates is \(\frac{-3y + 12y}{2} = \frac{9y}{2}\)

Thus, the midpoint M is at \(M(16y, \frac{9y}{2})\).
This formula is quite helpful in dividing segments into equal halves and is frequently used in graphical analysis and applications where symmetry and central points are assessed.

Coordinate Geometry

Coordinate Geometry, also known as Analytic Geometry, combines algebra and geometry to study geometric figures on a coordinate plane. Here are some key aspects:

• Allows the determination of distances, midpoints, and slopes of lines through algebraic formulas.
• Provides a way to solve geometrical problems numerically.
• Facilitates the visualization of algebraic equations as geometric figures, bridging abstract and concrete concepts.

Utilizing the Distance and Midpoint formulas are crucial to coordinate geometry. These tools help to calculate exact measurements such as the length of a line or the point that bisects a line segment. Understanding this means mastering the connection between algebraic expressions and geometrical interpretations.
By practicing with these formulas, students gain a deeper insight into how shapes and lines behave within a coordinate grid, building a foundation for more complex geometrical concepts and analytical methods.