Problem 33 - 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. \((3,4),(-3,-4)\)

Step-by-Step Solution

Answer

Plot the points (3,4) and (-3,-4). Distance is 10 units. Mid-point is (0,0).

Step 1: Plot the Points

Begin by plotting the two given points on a coordinate plane. The first point is (3, 4), which is located 3 units to the right of the origin on the x-axis and 4 units up on the y-axis. The second point is (-3, -4), which is 3 units to the left of the origin and 4 units down on the y-axis.

Step 2: Calculate the Distance Between Points

To find the distance between two points (3, 4)and(-3, -4), use the distance formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).Substitute the values: \( x_1 = 3 \), \( y_1 = 4 \), \( x_2 = -3 \), \( y_2 = -4 \).\[\text{Distance} = \sqrt{(-3 - 3)^2 + (-4 - 4)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100}\]The distance is 10 units.

Step 3: Find the Mid-point

To find the mid-point of the segment joining the points (3, 4) and (-3, -4), use the mid-point formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).Substitute the values: \( x_1 = 3 \), \( y_1 = 4 \), \( x_2 = -3 \), \( y_2 = -4 \).\[\text{Mid-point} = \left( \frac{3 + (-3)}{2}, \frac{4 + (-4)}{2} \right) = \left( 0, 0 \right)\]The mid-point is at the origin, (0, 0).

Key Concepts

Distance Formula

The distance formula is a fundamental tool in coordinate geometry. It allows us to determine the length of the line segment connecting two points in the plane. To calculate the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the formula used is:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here’s how it works:

• Subtract the x-coordinates (\(x_2 - x_1\)). This gives the difference in the horizontal (x-axis) direction.
• Subtract the y-coordinates (\(y_2 - y_1\)). This gives the difference in the vertical (y-axis) direction.
• Square both differences: \( (x_2 - x_1)^2 \) and \( (y_2 - y_1)^2 \) to remove any negative signs and because distance is always positive.
• Add these squares together.
• Finally, take the square root of this sum to get the distance between the two points.

This formula essentially uses the Pythagorean theorem to find the distance as a hypotenuse of a right-angled triangle formed by the horizontal and vertical differences. It's a straightforward but crucial technique for distance measurement in geometry.

Mid-point Formula

The mid-point formula helps find the exact middle point of the line segment joining two points. For the points \( (x_1, y_1) \) and \( (x_2, y_2) \), the mid-point is calculated by:\[ \text{Mid-point} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]This is what you do:

• Add the x-coordinates of the two points together (\(x_1 + x_2\)) and divide by 2. This gives the x-coordinate of the mid-point.
• Add the y-coordinates of the two points together (\(y_1 + y_2\)) and divide by 2. This gives the y-coordinate of the mid-point.

Imagine the mid-point as the "balance point" on the segment. It's an average of both x and y coordinates, marking the center of a straight line between the two points. This concept is incredibly useful, especially in geometry and coordinate plane problems where symmetry and central locations are critical.

Plotting Points

Plotting points on a coordinate plane is the foundation of graphing in math and is visually representing ordered pairs. When you have a point \( (x, y) \):

• The first number, \( x \), is the x-coordinate telling you how far to move horizontally from the origin.
• The second number, \( y \), is the y-coordinate which indicates how far to move vertically.

Coordinate planes consist of two perpendicular lines or axes. The horizontal axis is the x-axis, and the vertical is the y-axis. These two axes intersect at the origin, or \( (0,0) \), which is the starting point for plotting.You first locate the x-value on the x-axis - move right if it's positive, left if it's negative.Then, from there, find the y-value by moving up if it's positive or down if it's negative.Each point in coordinate geometry gives you a precise location that can be used for further calculations, such as finding distances or mid-points, as demonstrated in our above examples. Always plot points accurately to ensure correctness in subsequent geometric computations.