1.
A line in the standard (x, y) coordinate plane contains the points P(3, -5) and Q (-7, 9). What point is the midpoint of PQ?
Correct Answer
C. (-2, 2)
Explanation
The midpoint of a line segment is the point that is equidistant from both endpoints. To find the midpoint of PQ, we can use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Plugging in the coordinates of P(3, -5) and Q(-7, 9) into the formula, we get:
Midpoint = ((3 + (-7))/2, (-5 + 9)/2)
= (-4/2, 4/2)
= (-2, 2)
Therefore, the point (-2, 2) is the midpoint of PQ.
2.
Calculate the distance between (4, -3) and (-3, 8).
Correct Answer
A.
Explanation
To calculate the distance between two points in a coordinate plane, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by the square root of the sum of the squares of the differences in the x-coordinates and the y-coordinates of the two points. In this case, the x-coordinate differences are 4 - (-3) = 7 and the y-coordinate differences are (-3) - 8 = -11. Squaring these differences, we get 7^2 = 49 and (-11)^2 = 121. Adding these squares together, we get 49 + 121 = 170. Taking the square root of 170 gives us approximately 13.038, which is the distance between the two points.
3.
What is the midpoint of the line segment whose endpoints are (6, 0) and (3, 7)?
Correct Answer
C.
Explanation
The midpoint of a line segment is the point that is equidistant from both endpoints. To find the midpoint, we take the average of the x-coordinates and the average of the y-coordinates of the endpoints. In this case, the average of the x-coordinates is (6+3)/2 = 4.5 and the average of the y-coordinates is (0+7)/2 = 3.5. Therefore, the midpoint of the line segment is (4.5, 3.5).
4.
Find the distance between points A (9, 7) and B (1, 1).
Correct Answer
A. 10
Explanation
The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. The formula is √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points. Using this formula, the distance between point A (9, 7) and point B (1, 1) is √((1 - 9)^2 + (1 - 7)^2) = √((-8)^2 + (-6)^2) = √(64 + 36) = √100 = 10. Therefore, the correct answer is 10.
5.
Find the midpoint of the segment defined by points C (4, -5) and D (-8, 9).
Correct Answer
C. (-2, 2)
Explanation
The midpoint of a line segment is the average of the coordinates of its endpoints. To find the midpoint of the segment CD, we can average the x-coordinates and the y-coordinates separately. The average of the x-coordinates (4 and -8) is (-8+4)/2 = -4/2 = -2. The average of the y-coordinates (-5 and 9) is (-5+9)/2 = 4/2 = 2. Therefore, the midpoint of CD is (-2, 2).
6.
The coordinates of the Sears Tower are at (-1, -1) and Millenium Park is at (2, 2). What is the distance between the Sears Tower and Millenium Park?
Correct Answer
E.
Explanation
The distance between two points in a coordinate plane can be found using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is equal to the square root of [(x2 - x1)^2 + (y2 - y1)^2]. In this case, the coordinates of the Sears Tower are (-1, -1) and Millenium Park is (2, 2). Plugging these values into the distance formula, we get the square root of [(2 - (-1))^2 + (2 - (-1))^2], which simplifies to the square root of [9 + 9], or the square root of 18. The square root of 18 is approximately 4.24. Therefore, the distance between the Sears Tower and Millenium Park is approximately 4.24.
7.
Consider two stores in a town. The first store is a DVD rentals store. The second is a pizza place where customers collect their pizzas after they order pizza online. The DVD rentals store is represented by the coordinates (-4, 2) and the online pizza place is represented by the coordiantes (2, -4). If the DVD rental store and the online pizza place are connected by a line segment, what is the midpoint of this line?
Correct Answer
B. (-1, -1)
Explanation
The midpoint of a line segment is the point that is equidistant from both endpoints of the segment. In this case, the coordinates of the first store are (-4, 2) and the coordinates of the second store are (2, -4). To find the midpoint, we take the average of the x-coordinates and the average of the y-coordinates. The average of -4 and 2 is -1, and the average of 2 and -4 is -1. Therefore, the midpoint of the line segment connecting the two stores is (-1, -1).
8.
Find the distance, in coordinate units, between the points (1, -7) and (-2, 2) in the standard xy coordinate plane.
Correct Answer
E.
Explanation
The distance between two points in the coordinate plane can be found using the distance formula, which is the square root of the sum of the squares of the differences between the x and y coordinates of the two points. In this case, the x coordinates are 1 and -2, and the y coordinates are -7 and 2. The difference between the x coordinates is 1 - (-2) = 3, and the difference between the y coordinates is -7 - 2 = -9. Squaring these differences gives 3^2 = 9 and (-9)^2 = 81. Adding these squares gives 9 + 81 = 90. Taking the square root of 90 gives approximately 9.49, which is the distance between the two points.
9.
What is the distance between the two points: (3, 5) and (-2, 4)?
Correct Answer
A. 5.10
Explanation
The distance between two points can be calculated using the distance formula, which is derived from the Pythagorean theorem. The formula is: √((x2 - x1)^2 + (y2 - y1)^2). In this case, the coordinates are (3, 5) and (-2, 4). Plugging these values into the formula, we get: √((-2 - 3)^2 + (4 - 5)^2) = √((-5)^2 + (-1)^2) = √(25 + 1) = √26. Rounded to two decimal places, the answer is 5.10.
10.
What is the distance between the two points below?
Correct Answer
D. 3.61
Explanation
The given answer, 3.61, is the distance between the two points mentioned in the question. However, without any context or additional information, it is not possible to determine what these two points represent or how the distance is calculated. Therefore, a more specific explanation cannot be provided.
11.
What is the midpoint between the two points below?
Correct Answer
C. (0, 1.5)
Explanation
The midpoint between two points is the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of the midpoint is (2.5 + 2) / 2 = 2.25, and the y-coordinate of the midpoint is (-1 + 1.5) / 2 = 0.25. Therefore, the midpoint between the two points is (2.25, 0.25). However, this point is not one of the given options. Therefore, the correct answer cannot be determined based on the given options.
12.
What is the midpoint of the two points: (-4, 3) and (2, 5)?
Correct Answer
A. (-1, 4)
Explanation
The midpoint of two points is the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of the midpoint is (-4 + 2)/2 = -1 and the y-coordinate of the midpoint is (3 + 5)/2 = 4. Therefore, the midpoint of the two points (-4, 3) and (2, 5) is (-1, 4).
13.
What is the distance of line segment AB with A(-1, 3) and B(5, -5)?
Correct Answer
C. 10 Units
Explanation
The distance between two points in a Cartesian coordinate system can be calculated using the distance formula, which is derived from the Pythagorean theorem. The distance formula is: Distance = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we have point A(-1, 3) with coordinates (x1, y1) and point B(5, -5) with coordinates (x2, y2).
Plug these values into the distance formula:
Distance = √((5 - (-1))^2 + (-5 - 3)^2)
Simplify the equation:
Distance = √((5 + 1)^2 + (-5 - 3)^2) Distance = √(6^2 + (-8)^2)
Calculate the squares:
Distance = √(36 + 64)
Add the squares:
Distance = √(100)
Take the square root:
Distance = 10
So, the distance between point A and point B is 10 units.
14.
What is the midpoint of A(2, 3) and B(-2, 5)?
Correct Answer
D. (0,4)
Explanation
The midpoint of two points is found by taking the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of the midpoint is (2 + (-2))/2 = 0, and the y-coordinate is (3 + 5)/2 = 4. Therefore, the midpoint of A(2, 3) and B(-2, 5) is (0, 4).
15.
What is the distance of line segment AB with A(-1, 3) and B(5, -5)?
Correct Answer
C. C
Explanation
The distance of line segment AB can be calculated using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is equal to the square root of (x2 - x1)^2 + (y2 - y1)^2. In this case, the coordinates of point A are (-1, 3) and the coordinates of point B are (5, -5). Plugging these values into the distance formula, we get the distance of line segment AB as sqrt((5 - (-1))^2 + (-5 - 3)^2), which simplifies to sqrt(6^2 + (-8)^2), and further simplifies to sqrt(36 + 64), which is equal to sqrt(100). Therefore, the distance of line segment AB is 10 units.