Problem 23 - Precalculus

Question

Find the distance \(d\) between the points and the midpoint \(M\) of the line segment which connects them. $$ (3,-10),(-1,2) $$

Step-by-Step Solution

Answer

The distance is \(4\sqrt{10}\) and the midpoint is \((1, -4)\).

Step 1: Determine the change in x and y

Identify the coordinates of the two points, which are \((x_1, y_1) = (3, -10)\) and \((x_2, y_2) = (-1, 2)\). Compute the change in x (\(\Delta x\)) and change in y (\(\Delta y\)) using the formulas: \(\Delta x = x_2 - x_1\) and \(\Delta y = y_2 - y_1\). This gives us \(\Delta x = -1 - 3 = -4\) and \(\Delta y = 2 - (-10) = 12\).

Step 2: Use the distance formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). Substitute the values \((x_1, y_1) = (3, -10)\) and \((x_2, y_2) = (-1, 2)\) into the formula: \[d = \sqrt{(-4)^2 + (12)^2} = \sqrt{16 + 144} = \sqrt{160}\].

Step 3: Simplify the distance

Simplify \(\sqrt{160}\). Since \(160 = 16 \times 10\), rewrite \(\sqrt{160}\) as \(\sqrt{16 \times 10}\). This simplifies further as \(\sqrt{16} \times \sqrt{10} = 4\sqrt{10}\). Thus, the distance \(d\) is \(4\sqrt{10}\).

Step 4: Determine the midpoint

The midpoint \(M\) of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\), which calculates to \(M = \left( \frac{3 + (-1)}{2}, \frac{-10 + 2}{2} \right) = \left( 1, -4 \right)\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a critical part of mathematics that helps us visualize and analyze geometric shapes using algebra and the Cartesian plane. This branch of geometry enables us to use coordinates to represent points and lines in a two-dimensional space. Here’s how it all comes together:

• We place points on a grid known as the Cartesian plane. This plane has two axes: the horizontal x-axis and the vertical y-axis.
• Any point on this plane can be represented by a pair of numbers, known as coordinates \((x, y)\), where x represents the horizontal position and y the vertical position.
• Understanding coordinate geometry is essential for solving various mathematical problems, including calculating distances and midpoints between points.

Coordinate geometry, sometimes known as analytical geometry, is the foundation for more complex topics like trigonometry and calculus. It allows us to make sense of visual problems using numbers and equations.

Midpoint Formula

The midpoint formula is a helpful tool in coordinate geometry that allows us to find the exact center between two points on a line segment. Suppose you have two points \((x_1, y_1)\) and \((x_2, y_2)\). The formula you will use to find the midpoint \(M\) is:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]This formula gives you the average of the x-coordinates and the y-coordinates of the two points:

• The x-coordinate of the midpoint is the average of \(x_1\) and \(x_2\): \(\frac{x_1 + x_2}{2}\).
• The y-coordinate of the midpoint is the average of \(y_1\) and \(y_2\): \(\frac{y_1 + y_2}{2}\).

This straightforward calculation provides a point that divides the line segment into two equal halves. It's particularly useful in geometry and algebra, enabling us to locate the center of a line quickly and effectively, as exemplified in the solution where \((3, -10)\) and \((-1, 2)\) resulted in a midpoint of \((1, -4)\).

Distance Between Two Points

To find the distance between two points on a plane, we use the distance formula, which derives from the Pythagorean theorem. The formula is handy for measuring the length of the straight line that connects two points—\((x_1, y_1)\) and \((x_2, y_2)\):\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Here's a breakdown of how it works:

• Calculate the difference in the x-coordinates \(\Delta x\) by \((x_2 - x_1)\) and the difference in the y-coordinates \(\Delta y\) by \((y_2 - y_1)\).
• Square each difference to eliminate any negative signs and emphasize magnitude.
• Add these squared differences, then take the square root of the total to find the actual distance.

For example, using the points \((3, -10)\) and \((-1, 2)\), we found the distance to be \(4\sqrt{10}\). This formula is indispensable in both practical applications and theoretical geometry, providing a reliable method for measuring the exact distance between any two points on a coordinate plane.