SAT Math : Coordinate Geometry
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Example Questions
Example Question #13 : Midpoint Formula
The following coordinates represent the vertices of a box. Where does the center of the box lie?
None of the given answers
To solve this, we need to choose two points that lie diagonally across from each other. Let's use 
Example Question #14 : Midpoint Formula
Give the coordinates of the midpoint, in terms of 
The coordinates 
and
Setting 
Setting 
The coordinates of the midpoint, in terms of 
Example Question #12 : How To Find The Midpoint Of A Line Segment
The two endpoints of a line segment are 
In order to find the midpoint of a line segment, you need to average the x and y values of the endpoints.
The midpoint formula is
After plugging in the values you get
and 
Therefore, the midpoint is at 
Example Question #1 : X And Y Intercept
Solve the equation for x and y.
x² + y = 31
x + y = 11
x = 13, 7
y = 8, –6
x = 5, –4
y = 6, 15
x = 6, 15
y = 5, –4
x = 8, –6
y = 13, 7
x = 5, –4
y = 6, 15
Solving the equation follows the same system as the first problem. However since x is squared in this problem we will have two possible solutions for each unknown. Again substitute y=11-x and solve from there. Hence, x2+11-x=31. So x2-x=20. 5 squared is 25, minus 5 is 20. Now we know 5 is one of our solutions. Then we must solve for the second solution which is -4. -4 squared is 16 and 16 –(-4) is 20. The last step is to solve for y for the two possible solutions of x. We get 15 and 6. The graph below illustrates to solutions.
Example Question #2 : X And Y Intercept
Solve the equation for x and y.
x² – y = 96
x + y = 14
x = 15, 8
y = 5, –14
x = 5, –14
y = 15, 8
x = 10, –11
y = 25, 4
x = 25, 4
y = 10, –11
x = 10, –11
y = 25, 4
This problem is very similar to number 2. Derive y=14-x and solve from there. The graph below illustrates the solution.
Example Question #3 : X And Y Intercept
Solve the equation for x and y.
5x² + y = 20
x² + 2y = 10
x = 14, 5
y = 4, 6
x = √4/5, 7
y = √3/10, 4
x = √10/3, –√10/3
y = 10/3
No solution
x = √10/3, –√10/3
y = 10/3
The problem involves the same method used for the rest of the practice set. However since the x is squared we will have multiple solutions. Solve this one in the same way as number 2. However be careful to notice that the y value is the same for both x values. The graph below illustrates the solution.
Example Question #55 : Coordinate Geometry
Solve the equation for x and y.
x² + y = 60
x – y = 50
x = –40, –61
y = 10, –11
x = 10, –11
y = –40, –61
x = 11, –10
y = 40, 61
x = 40, 61
y = 11, –10
x = 10, –11
y = –40, –61
This is a system of equations problem with an x squared, to be solved just like the rest of the problem set. Two solutions are required due to the x2. The graph below illustrates those solutions.
Example Question #1 : X And Y Intercept
A line passes through the points 
None of the available answers
First we will calculate the slope as follows:
And our equation for a line is
Now we need to calculate b. We can pick either of the points given and solve for
Our equation for the line becomes
Example Question #1 : How To Find X Or Y Intercept
If the equation of a line is 4y – x = 48, at what point does that line cross the x-axis?
(0,–48)
(–48,0)
(0,–12)
(0,12)
(48,0)
(–48,0)
When the equation crosses the x-axis, y = 0. Plug 0 into the equation for y, and solve for x.
4(0) – x = 48, –x = 48, x = –48
Example Question #2 : How To Find X Or Y Intercept
The slope of a line is equal to -3/4. If that line intersects the y-axis at (0,15), at what point does it intersect the x-axis?
15
5
60
20
-20
20
If the slope of the line m=-3/4, when y=15 and x=0, plug everything into the equation y=mx+b.
Solving for b:
15=(-3/4)*0 + b
b=15
y=-3/4x + 15
To get the x-axis intersect, plug in y=0 and solve for x.
0 = -3/4x + 15
3/4x = 15
3x = 15*4
x = 60/3 = 20
x=20
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