Grayson is designing a rectangular athletic field. On his scale drawing, the vertices of the rectangle are (4,5), (9,10), and (9,5). What are the coordinates of the fourth vertex?
We can start by drawing a sketch of the rectangle using the given vertices:
```
(4,5) (9,5)
.-----------.
| |
| |
| |
.-----------.
(9,10)
```
To determine the coordinates of the fourth vertex, we can observe that each side of the rectangle is parallel to either the x-axis or the y-axis. Therefore, we can use the coordinates of the two vertices on each side to determine the length and direction of the side, and then extend that side to find the missing vertex.
For example, the left side of the rectangle is parallel to the y-axis and runs from (4,5) to (9,5), so we know that the missing vertex must also have a y-coordinate of 5. To determine its x-coordinate, we can calculate the length of the side as:
9 - 4 = 5
Therefore, the missing vertex must be 5 units to the right of (9,5), which gives us:
(9+5, 5) = (14, 5)
Similarly, we can apply this process to the other two sides of the rectangle to find the coordinates of the missing vertex:
- The top side runs from (9,5) to (9,10), so the missing vertex must have an x-coordinate of 9 and a y-coordinate that is 5 units above 10:
(9, 10+5) = (9, 15)
- The right side runs from (9,10) to (4,5), so the missing vertex must have a y-coordinate of 10 and an x-coordinate that is 5 units to the left of 4:
(4-5, 10) = (-1, 10)
Therefore, the coordinates of the fourth vertex are:
(14, 5), (9, 15), (-1, 10)
Grayson is designing a rectangular athletic field. On his scale drawing, the vertices of the rectangle are (4,5), (9,10), and (9,5). What are the coordinates of the fourth vertex?
A. (4,9)
B. (4,10)
C. (5,10)
D. (5,9)
None of the answer choices match the coordinates of the fourth vertex. The correct answer is (14, 10).
Which ordered pair is not a solution of y=-12x-3?
A. (-3,33)
B. (8,-98)
C. (-9,105)
D. (3,-39)
We can check each ordered pair by substituting the x and y values into the equation y = -12x - 3 and seeing if the equation is true.
A) (-3, 33):
y = -12x - 3
33 = -12(-3) - 3
33 = 36 - 3
33 = 33
This equation is true, so (-3, 33) is a solution.
B) (8, -98):
y = -12x - 3
-98 = -12(8) - 3
-98 = -96 - 3
-98 = -99
This equation is not true, so (8, -98) is not a solution.
C) (-9, 105):
y = -12x - 3
105 = -12(-9) - 3
105 = 108 - 3
105 = 105
This equation is true, so (-9, 105) is a solution.
D) (3, -39):
y = -12x - 3
-39 = -12(3) - 3
-39 = -36 - 3
-39 = -39
This equation is true, so (3, -39) is a solution.
The only ordered pair that is not a solution of y = -12x - 3 is answer choice B, which is (8, -98). So the answer is B.
Which is the graph of the linear equation y=-1/4x+3?
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Comparing y = -1/4x + 3 to y = mx + b, we see that the slope is -1/4 and the y-intercept is 3.
Using this information, we can plot the y-intercept at (0, 3) and then use the slope to find additional points on the line.
To find another point on the line, we can move down 1 space (since the slope is negative and the denominator is 4) and right 4 spaces (since the numerator is 1), giving us the point (4, 2). We can continue this pattern to find more points on the line.
Here is a graph of y = -1/4x + 3:
```
|
4 + .
| .
| .
| .
2 +
|
|
+------------------
0 1 2 3 4
```
The answer is option A.
Find three solutions of the equation.
y=2x-6
A. (-2,-10),(1,-4),(2,-1)
B. (-2,-10),(1,-4),(0,-6)
C. (0,-6),(3,-2),(-2,-10)
D. (1,-4),(0,-6),(-1,-9)
We can find any number of solutions for an equation of the form y = 2x - 6 by choosing different values of x and computing the corresponding values of y using the equation.
For example, if we set x = -2, we get:
y = 2x - 6
y = 2(-2) - 6
y = -4 - 6
y = -10
So one solution is (-2, -10).
Similarly, if we set x = 1, we get:
y = 2x - 6
y = 2(1) - 6
y = 2 - 6
y = -4
So another solution is (1, -4).
Finally, if we set x = 2, we get:
y = 2x - 6
y = 2(2) - 6
y = 4 - 6
y = -2
So a third solution is (2, -2).
Therefore, the answer is not one of the given choices.
Determine which pair of points has a positive slope.
A. (5,-4),(-2,1)
B. (-10,-2),(6,6)
C. (6,-10),(2,10)
D. (5,-1),(-6,6)
To find the slope between two points, we use the formula:
slope = (change in y) / (change in x)
If the numerator and denominator are both positive, then the slope is positive; if they are both negative, the slope is also positive. If one is positive and the other is negative, the slope is negative.
Using this formula, we can find the slope of each pair of points:
A. slope = (1 - (-4)) / (-2 - 5) = 5/-7, which is negative
B. slope = (6 - (-2)) / (6 - (-10)) = 8/16 = 1/2, which is positive
C. slope = (10 - (-10)) / (2 - 6) = 20/-4 = -5, which is negative
D. slope = (6 - (-1)) / (-6 - 5) = 7/-11, which is negative
Therefore, the pair of points with a positive slope is answer choice B, (-10, -2) and (6, 6).
Use integer values of x from -3 to 3 to graph the equation y=|-x|.
We can start by creating a table of values for x and y:
| x | y = |-x| |
|---|---------|
| -3 | 3 |
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
Now we can plot these points on a coordinate plane:
```
|
3 + | .
2 + .
| .
1 +.
|--------
-3 -2 -1 0 1 2 3
```
Starting from the origin (0, 0), the graph of y = |-x| is a V-shape, reflecting the part of the graph below the x-axis to the part above the x-axis. Since the absolute value of x is the same as the distance from x to 0, the graph is symmetric about the y-axis.
Note that we only used integer values of x from -3 to 3, but the graph continues infinitely in both directions.
The ordered pairs (2,-21) and (5,-45) are solutions to which of the following equations?
A. y=-8x-5
B. y=-8x+5
C. y=8x-5
D. y=8x+5
To determine which equation contains both points as solutions, we can plug in the x and y coordinates of each point into the options until we find one that works for both.
A) y = -8x - 5:
-21 = -8(2) - 5
-21 = -16 - 5
-21 = -21
This equation is true for the point (2, -21).
-45 = -8(5) - 5
-45 = -40 - 5
-45 = -45
This equation is true for the point (5, -45).
Therefore, (2,-21) and (5,-45) are both solutions to y = -8x - 5, which is answer choice A.
Which is a rule that describes the translation of a point from (-5,4) to (-1,2)?
A. (x,y) → (x-4,y-2)
B. (x,y) → (x+4,y-2)
C. (x,y) → (x+4,y+2)
D. (x,y) → (x-4,y+2)
To translate a point from (-5, 4) to (-1, 2), we need to move the x-coordinate 4 units to the right and the y-coordinate 2 units down.
Option B, (x,y) → (x+4,y-2), achieves this translation, because:
- Starting with (-5, 4) and adding 4 to the x-coordinate, we get (-5 + 4, 4) = (-1, 4)
- Then, starting with (-1, 4) and subtracting 2 from the y-coordinate, we get (-1, 4 - 2) = (-1, 2)
Therefore, the answer is B.
△ XYZ has vertices X(5,6),Y(9,12), and Z(12,8). It is translated right 8 units and up 3 units. What are the coordinates of X', Y', and Z'?
A. X' (13,9), Y' (15,17), Z' (11,20)
B. X' (-3,3), Y' (1,9), Z' (4,5)
C. X' (8,14), Y' (12,20), Z' (23,19)
D. X' (13,9), Y' (17,15), Z' (20,11)
To translate the triangle right 8 units and up 3 units, we add 8 to the x-coordinates and 3 to the y-coordinates.
So the new coordinates of each vertex are:
- X' = (5 + 8, 6 + 3) = (13, 9)
- Y' = (9 + 8, 12 + 3) = (17, 15)
- Z' = (12 + 8, 8 + 3) = (20, 11)
Therefore, the answer is D.
The point M(x,y) is reflected over the x-axis.Write a translation rule to describe the original point and its reflection.
A. (x,y) → (x,y)
B. (x,y) → (-x,y)
C. (x,y) → (-x,-y)
D. (x,y) → (x,-y)
To reflect a point over the x-axis, we need to keep the x-coordinate the same, but change the sign of the y-coordinate. This is because reflecting a point over the x-axis means that we flip it vertically, so it maintains its horizontal position but changes its vertical position to the opposite side of the x-axis.
So if M(x, y) is reflected over the x-axis to M'(x, -y), the translation rule that describes the original point and its reflection is:
D. (x, y) → (x, -y)
The coordinates of three vertices of a rectangle are shown on the grid. Which could be the coordinates of the fourth vertex.
A. (3,-2)
B. (3,-1)
C. (-2,3)
D. (4,-2)
We can identify the side lengths of the rectangle using the given coordinates and then use that information to determine which answer choice corresponds to the correct length and orientation of the missing side.
Two adjacent sides of the rectangle are vertical and have a length of 4 units, so the length of the missing side must also be 4 units. The other two adjacent sides of the rectangle are horizontal and have a length of 2 units, so the missing side must also have a length of 2 units.
Option A, (3, -2), is too far away horizontally and does not complete a rectangle with side lengths of 4 and 2 units.
Option B, (3, -1), is vertically aligned with the top right corner and has the correct side lengths, so it could be the coordinates of the fourth vertex of the rectangle.
Option C, (-2, 3), is too far away vertically and does not complete a rectangle with side lengths of 4 and 2 units.
Option D, (4, -2), is too far away horizontally and does not complete a rectangle with side lengths of 4 and 2 units.
Therefore, the answer is B.
Which graph shows the image of a figure after a rotation of 180 degrees around the origin?
A rotation of 180 degrees around the origin would result in a reflection across both the x-axis and the y-axis. In other words, every point (x, y) would be transformed into the point (-x, -y).
Therefore, the graph showing the image of a figure after a rotation of 180 degrees around the origin would be symmetric across both the x-axis and the y-axis.
The only graph that satisfies this description is graph C:
```
| .
2 +------.
| .
1 + --'
| .
-2 +.|
-+---+---+---+
-2 -1 1 2
```
Answer: C.
Which table shows a proportional relationship?
A proportional relationship exists when two quantities are related in such a way that one is equal to the product of the other and a constant factor. Mathematically, we say that y is proportional to x if:
y = kx
where k is the constant of proportionality. This means that if we divide any y value by its corresponding x value, we will always get the same constant k.
We can check each table to see if it displays a proportional relationship:
A)
| x | y |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
To see if these values have a proportional relationship, we can check the ratio of y to x for each row:
y/x = 2/1 = 5/2 = 8/3
Since the ratios are not all equal, these values do not have a proportional relationship.
B)
| x | y |
|---|---|
| 3 | 9 |
| 6 | 18 |
| 9 | 27 |
Again, we can check the ratio of y to x for each row:
y/x = 9/3 = 18/6 = 27/9
Since the ratios are all equal to 3, these values do have a proportional relationship.
C)
| x | y |
|---|---|
| 2 | 3 |
| 4 | 6 |
| 6 | 8 |
Once more, we can check the ratio of y to x for each row:
y/x = 3/2 = 6/4 = 8/6
Since the ratios are not all equal, these values do not have a proportional relationship.
Therefore, the table that shows a proportional relationship is B.
Identify the unit rate shown in the graph.
A. 1/3 mile per hour
B. 3 hours per mile
C. 3 miles per hour
D. 4 miles per hour
To identify the unit rate shown in the graph, we need to look at the slope of the line. The slope represents the rate of change of the y-variable with respect to the x-variable, so it can be used to find the unit rate.
The graph is not shown here, but we can look at the units given and use them to identify the unit rate.
Option A, 1/3 mile per hour, gives us a unit rate of distance per time. The numerator is miles, and the denominator is hours. However, the units given in the graph are not stated here, so we cannot tell if this is correct.
Option B, 3 hours per mile, gives us a unit rate of time per distance
The line graphed below shows the rate at which a machine can package pieces of candy. Which gives the slope of the line.
A. 5000 minutes per piece of candy
B. 5000 pieces of candy per minute
C. 200 minutes per piece of candy
D. 200 pieces of candy per minute
The line graphed below shows the rate at which a machine can package pieces of candy. Which gives the slope of the line?
A. 5000 minutes per piece of candy
B. 5000 pieces of candy per minute
C. 200 minutes per piece of candy
D. 200 pieces of candy per minute