# Coordinate Geometry Hard

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Difficult coordinate geometry ACT questions

• 1.

### Given the vertices of parallelogram QRST in the standard (x, y) coordinate plane below, what is the area of triangle QRS, in square units?

• A.

24

• B.

28

• C.

48

• D.

60

• E.

80

A. 24
Explanation
Area triangle = 1/2bh and the base is 6 units (from -3 to 3) and the height is 8 units (from -5 to 3). 1/2*6*8= 24 sq units

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• 2.

### In the standard (x, y) coordinate plane, what is the equation of the line that passes through the origin and the point (3, 4)?

• A.

Y = 1/4x + 3/4

• B.

Y = 1/4x - 1/3

• C.

Y = 4/3x

• D.

Y = 1/2x + 3/4

• E.

Y = 9/4x

C. Y = 4/3x
Explanation
slope = 3/4. remember the origin is just the point (0, 0) so you can use the slope formula followed by point slope form: y - y1 = m(x - x1) but if you think about it, b has to be 0 if it goes through the origin!

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• 3.

### For what value of n would the following system of equations have an infinite number of solutions?3a + b = 1212a + 4b = 3n

• A.

4

• B.

9

• C.

16

• D.

36

• E.

48

C. 16
Explanation
For infinite solutions, every term in the second equation would have to be a multiple of each term in the first equation. The a and b terms are clearly multiplied by 4. 12*4 = 48. So your equation is 48 = 3n, solve for n.

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• 4.

### What is the slope of the line that is perpendicular to the line determined by the equation 7x + 4y = 11?

• A.

-4

• B.

-7/4

• C.

11/4

• D.

4

• E.

4/7

E. 4/7
Explanation
Solve for y - then take the opposite reciprocal of the slope since the line is perpendicular

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• 5.

### What is the distance in the standard (x, y) coordinate plane between points (0, 1) and (4, 4)?

• A.

A

• B.

B

• C.

C

• D.

D

• E.

E

E. E
Explanation

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• 6.

### In the standard (x, y) coordinate plane, if the x-coordinate of each point on a line is 5 more than half the y-coordinate, what is the slope of the line?

• A.

-5

• B.

-1/2

• C.

1/2

• D.

2

• E.

5

D. 2
Explanation
x = 1/2y + 5. Solve for y and you get y = 2x - 5. This one is hard!

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• 7.

### A system of linear equations is shown below.4y = 3x + 12-4y = -3x - 8Which of the following describes the graph of this system of linear equations in the standard (x, y) coordinate plane?

• A.

Two parallel lines with negative slope

• B.

Two parallel lines with positive slope

• C.

A single line with negative slope

• D.

A single line with positive slope

• E.

Two perpendicular lines

B. Two parallel lines with positive slope
Explanation
The solution is no solution (0 = 4) so the lines are parallel and the slope of each of them is positive 3/4 (in the second one, the two negatives cancel).

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• 8.

### In the standard (x, y) coordinate plane, if a square has the vertices (-2, -3), (2, -3) and (2, 1), what is the set of coordinates for the final vertex?

• A.

(2, -1)

• B.

(1, -2)

• C.

(-1, 2)

• D.

(-2, -1)

• E.

(-2, 1)

E. (-2, 1)
Explanation
You just need to sketch this one to see it - don't try and figure it out in your head.

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• 9.

### Given that y - 5 = 1/2x + 1 is the equation of a line, at what point does it cross the x-axis?

• A.

-15

• B.

-12

• C.

1

• D.

4

• E.

6

B. -12
Explanation
It crosses the x-axis when y = 0, so 0-5 = 1/2x + 1. solve for x and you should get the right answer.

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• Current Version
• Mar 22, 2023
Quiz Edited by
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• Jun 23, 2010
Quiz Created by
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