Problem 28 - Algebra and Trigonometry

Question

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((-1,-1),(9,9)\)

Step-by-Step Solution

Answer

Distance: \(10\sqrt{2}\), Midpoint: \((4, 4)\).

Step 1: Plot the Points

To begin, plot the points \((-1, -1)\) and \((9, 9)\) on a coordinate plane. Point \((-1, -1)\) is found in the third quadrant, while \((9, 9)\) is in the first quadrant. Draw the points on the grid and connect them with a straight line to form the segment connecting the two points.

Step 2: Use the Distance Formula

To find the distance between the two points, use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plug in the coordinates \((-1, -1)\) and \((9, 9)\): \[ d = \sqrt{((9) - (-1))^2 + ((9) - (-1))^2} \] Calculate inside the parentheses: \[ d = \sqrt{(10)^2 + (10)^2} \] Then calculate the squares: \[ d = \sqrt{100 + 100} \] Finally, calculate the sum and the square root: \[ d = \sqrt{200} = 10\sqrt{2} \] Thus, the distance between the points is \(10\sqrt{2}\).

Step 3: Use the Midpoint Formula

To find the mid-point, use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substitute the given points :: \[ M = \left( \frac{-1 + 9}{2}, \frac{-1 + 9}{2} \right) \] Simplify inside the parentheses:\[ M = \left( \frac{8}{2}, \frac{8}{2} \right) \] Calculate the division to find the mid-point:\[ M = (4, 4) \] So, the midpoint of the line segment is \((4, 4)\).

Key Concepts

Distance Formula

The distance formula is a key tool in coordinate geometry. It helps us find the distance between two points in a coordinate plane using their coordinates. The points are usually represented as

• \((x_1, y_1)\)
• \((x_2, y_2)\)

To derive the distance, imagine forming a right triangle where the

• horizontal side is the difference in x-values: \((x_2 - x_1)\)
• vertical side is the difference in y-values: \((y_2 - y_1)\)

Using the Pythagorean Theorem, the distance \(d\) between the points is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In our problem, with points

• \((-1, -1)\) (in the third quadrant)
• \((9, 9)\) (in the first quadrant)

The calculation goes: \[ d = \sqrt{((9)-(-1))^2 + ((9)-(-1))^2} = \sqrt{200} = 10\sqrt{2} \]This gives us the distance as \(10\sqrt{2}\), showing the length of the segment between the two points.

Midpoint Formula

The midpoint formula is used to find the point that divides a line segment into two equal parts. This midpoint is average of the x-coordinates and y-coordinates of the points. If you have two points:

• Point one: \((x_1, y_1)\)
• Point two: \((x_2, y_2)\)

The formula is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] The midpoint, \(M\), lies directly in between the two points on the line segment. For our given points:

• \((-1, -1)\)
• \((9, 9)\)

We calculate the midpoint as follows:\[ M = \left( \frac{-1 + 9}{2}, \frac{-1 + 9}{2} \right) = (4, 4) \]This shows that the midpoint of the segment joining the points is \((4, 4)\). It acts as a balance point, ensuring both halves of the segment are equal in length.

Plotting Points

Plotting points is the foundational step in coordinate geometry, allowing you to visually represent equations and relationships on a graph. Each point is defined by an ordered pair \((x, y)\):

• The x-coordinate tells you the horizontal position on the x-axis
• The y-coordinate tells you the vertical position on the y-axis

To plot a point, start from the origin \((0, 0)\):

• Move along the x-axis according to the x-coordinate
• Move parallel to the y-axis according to the y-coordinate

For example, the point \((-1, -1)\) is located by moving one unit to the left and one unit down from the origin. The point \((9, 9)\) is nine units to the right and nine units up. When these points are joined with a straight line, the resulting segment can be analyzed for various properties, like distance and midpoint, using the corresponding formulas.