Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Find (a) the distance between \(P\) and \(Q\) and (b) the coordinates of the midpoint \(M\) of the segment joining \(P\) and \(Q\). $$P(-6,-10), Q(6,5)$$

Step-by-Step Solution

Answer

(a) Distance: 19.21; (b) Midpoint: (0, -2.5).

Step 1: Formula for Distance Between Two Points

To find the distance between two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), we use the distance formula: \ \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] For the points \(P(-6, -10)\) and \(Q(6, 5)\), identify \(x_1 = -6\), \(y_1 = -10\), \(x_2 = 6\), and \(y_2 = 5\).

Step 2: Plug Values into Distance Formula

Substitute the given values into the distance formula: \ \[d = \sqrt{(6 - (-6))^2 + (5 - (-10))^2}\] Simplify the expressions inside the parentheses: \ \[d = \sqrt{(6 + 6)^2 + (5 + 10)^2}\] which further simplifies to: \ \[d = \sqrt{12^2 + 15^2}\]

Step 3: Calculate the Distance

Calculate the squares: \ \[d = \sqrt{144 + 225}\] Then add the squares: \ \[d = \sqrt{369}\] Now, find the square root: \ \[d \approx 19.21\] So, the distance between the points \(P\) and \(Q\) is approximately \(19.21\).

Step 4: Formula for the Midpoint Coordinates

The formula for finding the midpoint \(M\) of a segment between points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is: \ \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\].

Step 5: Plug Values into Midpoint Formula

Substitute the given values into the midpoint formula: \ \[M = \left(\frac{-6 + 6}{2}, \frac{-10 + 5}{2}\right)\] Simplify the expressions: \ \[M = \left(0, -\frac{5}{2}\right)\] So, the coordinates of the midpoint \( M \) are \( (0, -2.5) \).

Key Concepts

Midpoint Formula

The midpoint formula helps you find the exact middle location between two points in a coordinate plane. It's like finding a balance point between two locations. The formula is quite simple:

• To find the midpoint, you take the average of the x-coordinates and the y-coordinates. This is expressed mathematically as \[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).\]
• Practical example: if you have points \(P(-6, -10)\) and \(Q(6, 5)\), plug these into the formula to get\(M = \left(\frac{-6+6}{2}, \frac{-10+5}{2}\right)\). This simplifies to \((0, -2.5)\).

This value gives you the exact midpoint, which is useful in various applications such as finding lines of symmetry or bisecting a shape.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate plane to visually represent spatial relationships.

• This branch of geometry uses coordinates (x, y) to not only show the location of points but also allows calculation of distances and midpoints, hence linking algebra with geometry.
• Coordinates make it easy to conceptually handle geometric shapes. With just the coordinates of crucial points, you can understand how to construct and define geometrical ideas.

Coordinate geometry is powerful in solving real-world problems, like navigation, where precise location tracking is needed.

Distance Calculation

Calculating the distance between two points on a plane is foundational in coordinate geometry. This is done using the distance formula, which is derived from the Pythagorean theorem.For any points \((x_1, y_1)\)and \((x_2, y_2)\), you can find the distance, \(d\),by plugging the coordinates into the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\]For our points \(P(-6, -10)\) and \(Q(6, 5)\), the calculated distance is\(\sqrt{369}\),approximately\(19.21\).This formula helps solve practical problems that require knowledge of lengths or sizes, such as engineering and design.

Midpoint Calculation

Calculating the midpoint involves finding the average of the two endpoints' coordinates. This straightforward process is made simple with the midpoint formula.

• For example, with endpoints \((-6, -10)\) and \((6, 5)\),calculate the x-coordinate by averaging \(-6\)and\(6\).
• Do the same for the y-coordinates: \(-10\)and \(5\).
• This gives: \(M = \left(\frac{-6 + 6}{2}, \frac{-10 + 5}{2}\right) = (0, -2.5)\).

The midpoint's use extends beyond simple calculation—it can help in dividing shapes, determining symmetry, or even predicting trends in data visualization. The midpoint is a crucial concept in both theoretical and applied mathematics.