Problem 48
Question
In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \( (1, 12) \), \( (6, 0) \)
Step-by-Step Solution
Verified Answer
The distance between points \( (1, 12) \) and \( (6, 0) \) is calculated to be \( \sqrt{149} \) and the midpoint of the line joining the points is \( (3.5, 6) \).
1Step 1: Plot the Points
Plot the points \( (1, 12) \) and \( (6, 0) \) on a standard set of axes. This will visually show where the points are located.
2Step 2: Calculate Distance
Utilize 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. Plug in \( x_1=1, y_1=12, x_2=6, y_2=0 \) into the formula and compute the distance.
3Step 3: Determine the Midpoint
Use the midpoint formula: \( M=\left( \frac{{x_1 +x_2}}{2}, \frac{{y_1+y_2}}{2}\right) \) where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points. Substitute \( x_1=1, y_1=12, x_2=6, y_2=0 \) into the formula and compute the midpoint.
Key Concepts
Distance FormulaMidpoint FormulaPlotting Points
Distance Formula
When studying coordinate geometry, the distance formula is an essential tool used for determining the length of a line segment connecting two points on a cartesian plane. The formula is rooted in the Pythagorean theorem and is expressed as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, the coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \) represent the respective positions of the two points on the grid. To apply the formula, first, identify the coordinates of the two points. Then, calculate the difference between the x-coordinates and the y-coordinates. Square these differences and add them together. Finally, take the square root of this sum to find the distance.
For instance, in our exercise involving points \( (1, 12) \) and \( (6, 0) \) we compute the distance as follows: the difference in the x-coordinates is \( 6 - 1 = 5 \) and the difference in the y-coordinates is \( 0 - 12 = -12 \) (which is 12 when squared, as negative signs are eliminated). The sum of the squares is \( 5^2 + 12^2 = 25 + 144 = 169 \) and the square root of 169 is 13, giving us a distance of 13 units. This method allows for a precise and direct calculation of the distance between any two points on a plane.
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, the coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \) represent the respective positions of the two points on the grid. To apply the formula, first, identify the coordinates of the two points. Then, calculate the difference between the x-coordinates and the y-coordinates. Square these differences and add them together. Finally, take the square root of this sum to find the distance.
For instance, in our exercise involving points \( (1, 12) \) and \( (6, 0) \) we compute the distance as follows: the difference in the x-coordinates is \( 6 - 1 = 5 \) and the difference in the y-coordinates is \( 0 - 12 = -12 \) (which is 12 when squared, as negative signs are eliminated). The sum of the squares is \( 5^2 + 12^2 = 25 + 144 = 169 \) and the square root of 169 is 13, giving us a distance of 13 units. This method allows for a precise and direct calculation of the distance between any two points on a plane.
Midpoint Formula
The midpoint of a line segment is the point that divides the segment into two equal parts. The midpoint formula comes into play when one needs to find the exact position of this dividing point. The formula is straightforward and is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
The \(x_1, y_1\) and \(x_2, y_2\) again are the coordinates of the endpoints of the line segment. To find the midpoint coordinates, simply add the x-coordinates of the endpoints and divide by 2 to find the x-coordinate of the midpoint. Repeat this process with the y-coordinates to determine the y-coordinate of the midpoint.
Applying it to our exercise, with endpoints \( (1, 12) \) and \( (6, 0) \), we calculate the midpoint by averaging the x-coordinates (1 and 6) to get 3.5, and the y-coordinates (12 and 0) to get 6. Therefore, the midpoint \( M \) of the line segment is \( (3.5, 6) \). By finding the midpoint, we essentially locate the 'center of mass' of a line segment, which can be invaluable for various geometric applications and proofs.
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
The \(x_1, y_1\) and \(x_2, y_2\) again are the coordinates of the endpoints of the line segment. To find the midpoint coordinates, simply add the x-coordinates of the endpoints and divide by 2 to find the x-coordinate of the midpoint. Repeat this process with the y-coordinates to determine the y-coordinate of the midpoint.
Applying it to our exercise, with endpoints \( (1, 12) \) and \( (6, 0) \), we calculate the midpoint by averaging the x-coordinates (1 and 6) to get 3.5, and the y-coordinates (12 and 0) to get 6. Therefore, the midpoint \( M \) of the line segment is \( (3.5, 6) \). By finding the midpoint, we essentially locate the 'center of mass' of a line segment, which can be invaluable for various geometric applications and proofs.
Plotting Points
Plotting points on a coordinate plane is a fundamental skill in geometry that provides a visual understanding of the position and relationships between different points. The coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point is determined by an ordered pair of numbers, referred to as coordinates. The first number corresponds to the x-axis (horizontal position), and the second number corresponds to the y-axis (vertical position).
To plot a point, start at the origin (0,0), where the x-axis and y-axis intersect. Move horizontally to reach the value of the x-coordinate and then vertically to reach the value of the y-coordinate. For example, for point \( (1, 12) \), move 1 unit right (positive direction on the x-axis) and then 12 units up (positive direction on the y-axis). Similarly, for point \( (6, 0) \), move 6 units right and stay on the x-axis since the y-coordinate is 0. Marking these points on graph paper or using digital graphing tools helps illustrate where they are located in relation to one another, and setting the stage for further calculations like the distance and midpoint.
To plot a point, start at the origin (0,0), where the x-axis and y-axis intersect. Move horizontally to reach the value of the x-coordinate and then vertically to reach the value of the y-coordinate. For example, for point \( (1, 12) \), move 1 unit right (positive direction on the x-axis) and then 12 units up (positive direction on the y-axis). Similarly, for point \( (6, 0) \), move 6 units right and stay on the x-axis since the y-coordinate is 0. Marking these points on graph paper or using digital graphing tools helps illustrate where they are located in relation to one another, and setting the stage for further calculations like the distance and midpoint.
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