Problem 132 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment connecting the points. $$(-4,-3),(6,10)$$

Step-by-Step Solution

Answer

The distance between points (-4,-3) and (6,10) is \(\sqrt{269}\) units and the midpoint of a line segment connecting these points is (1, 7/2).

Step 1: Plot the Points

The first task is to plot the points on the coordinate grid. The point \((-4,-3)\) is at 4 units left on x-axis and 3 units down on the y-axis. The point \((6,10)\) is at 6 units right on the x-axis and 10 units upward on the y-axis.

Step 2: Calculate the Distance

To find the distance between the two points use the distance formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\] where \((x_1,y_1)\) and \((x_2,y_2)\) are the coordinates of two points. Substitute \(-4\) for \(x_1\), \(-3\) for \(y_1\), \(6\) for \(x_2\), and \(10\) for \(y_2\). The calculation will be: \[d= \sqrt{(6-(-4))^2 + (10-(-3))^2} = \sqrt{(10)^2 + (13)^2} = \sqrt{100+169}= \sqrt{269}\]

Step 3: Calculate the Midpoint

To find the midpoint of the line segment connecting the two points, use the midpoint formula: \[( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} ) \] Substitute \(-4\) for \(x_1\), \(-3\) for \(y_1\), \(6\) for \(x_2\), and \(10\) for \(y_2\). The calculation will be: \[Midpoint = \left ( \frac{-4+6}{2}, \frac{-3+10}{2} \right ) = \left ( 1, \frac{7}{2} \right )\]

Key Concepts

Plotting Points

The first step in any coordinate geometry problem is plotting the points on a grid. This involves using the x-axis and y-axis to locate each point accurately.

For the point \((-4,-3)\), move 4 units to the left from the origin on the x-axis (because of the negative sign) and 3 units down on the y-axis. Conversely, the point \(6,10)\) requires you to move 6 units to the right on the x-axis and 10 units up on the y-axis.

Think of it as moving along a map: x indicates horizontal movement while y indicates vertical movement. Consistently apply this logic to plot any point, and visualize them on the grid for a clear mental picture.

Distance Formula

Finding the distance between two points in the coordinate plane requires using the distance formula. This formula expresses the Pythagorean Theorem to calculate the straight-line distance. Given by the formula:

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

Substitute the values from your points into the formula. For \((-4,-3)\) and \(6,10)\), this looks like:

- Substitute \(x_1 = -4\), \(y_1 = -3\), \(x_2 = 6\), and \(y_2 = 10\).
- Calculate: \(d= \sqrt{(6-(-4))^2 + (10-(-3))^2}\).
- Simplify to get \(d= \sqrt{269}\).

This value represents the exact length of the line segment connecting the two points, without needing to draw it physically.

Midpoint Formula

The midpoint formula helps you find the exact center of the line segment connecting two given points. It provides a quick and easy way to determine this point using:

\[( \frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} )\]

Use the coordinates of your points for substitution just as in the distance formula:

- Using \(x_1 = -4\), \(y_1 = -3\), \(x_2 = 6\), \(y_2 = 10\), plug these in:
- \(Midpoint = \left ( \frac{-4+6}{2}, \frac{-3+10}{2} \right )\)
- Simplify it to find the midpoint: \( \left ( 1, \frac{7}{2} \right )\).

The midpoint divides the segment into two equal parts and is very helpful in various geometric assessments.