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SSAT Upper Level Math : Coordinate Geometry

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Example Questions

Example Question #2 : How To Find Out If A Point Is On A Line With An Equation

Consider the lines described by the following two equations:

4y = 3x²

3y = 4x²

Find the vertical distance between the two lines at the points where x = 6.

Possible Answers:

Correct answer:

Explanation:

Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:

Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.

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Example Question #3 : Other Lines

For the line

$y=\frac{1}{3}x-7$

Which one of these coordinates can be found on the line?

Possible Answers:

(3, –6)

(9, 5)

(6, –12)

(6, 5)

(3, 7)

Correct answer:

(3, –6)

Explanation:

To test the coordinates, plug the x-coordinate into the line equation and solve for y.

y = 1/3x -7

Test (3,-6)

y = 1/3(3) – 7 = 1 – 7 = -6 YES!

Test (3,7)

y = 1/3(3) – 7 = 1 – 7 = -6 NO

Test (6,-12)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (6,5)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (9,5)

y = 1/3(9) – 7 = 3 – 7 = -4 NO

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Example Question #52 : Lines

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Possible Answers:

(3, 5)

(2, 2)

(3, –2)

(–2, –2)

(–2, 2)

Correct answer:

(–2, 2)

Explanation:

Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)

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Example Question #51 : Coordinate Geometry

Which of the following sets of coordinates are on the line $y=3x-4$ ?

Possible Answers:

$(2,2)$

$(1,5)$

$(3,4)$

$(1,2)$

$(2,-2)$

Correct answer:

$(2,2)$

Explanation:

$(2,2)$ when plugged in for $y$ and $x$ make the linear equation true, therefore those coordinates fall on that line.

$y=3x-4$

2=3(2)-4

2=6-4

2=2

Because this equation is true, the point must lie on the line. The other given answer choices do not result in true equalities.

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Example Question #52 : Coordinate Geometry

Which of the following points can be found on the line $\small y=3x+2$ ?

Possible Answers:

(-1,2)

(1, 5)

(2, 7)

(0, 1)

(1, 0)

Correct answer:

(1, 5)

Explanation:

We are looking for an ordered pair that makes the given equation true. To solve, plug in the various answer choices to find the true equality.

y=3x+2

5=3(1)+2

5=3+2

5=5

Because this equality is true, we can conclude that the point (1,5) lies on this line. None of the other given answer options will result in a true equality.

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Example Question #1 : How To Find Out If Lines Are Perpendicular

Two perpendicular lines intersect at the point (5,4) . One line passes through point (3,1) ; the other passes through point $\left ( 4, Y\right )$ . Evaluate .

Possible Answers:

$Y = 2\frac{1}{2}$

$Y =- 2\frac{1}{2}$

$Y = 5\frac{1}{2}$

$Y = 4\frac{2}{3}$

$Y = 3\frac{1}{3}$

Correct answer:

$Y = 4\frac{2}{3}$

Explanation:

The line that passes through $\left ( 3,1\right )$ and $\left ( 5,4\right )$ has slope

$m = \frac{4-1}{5-3} = \frac{3}{2}$ .

The line that passes through $\left ( 4, Y\right )$ and (5,4) , being perpendicular to the first, has as its slope the opposite reciprocal of $\frac{3}{2}$ , or $-\frac{2}{3}$ .

Therefore, to find , we use the slope formula and solve for :

$-\frac{2}{3} = \frac{Y-4}{4-5}$

$-\frac{2}{3} = \frac{Y-4}{-1}$

$-\frac{2}{3} \cdot (-1) = \frac{Y-4}{-1} \cdot (-1)$

$\frac{2}{3}= Y-4$

$\frac{2}{3}+ 4= Y-4 + 4$

$4\frac{2}{3}= Y$

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Example Question #111 : Geometry

A line has the following equation:

3y+12x=9

Which of the following could be a line that is perpendicular to this given line?

Possible Answers:

9y-2x=6

3y+9x=20

3x+8y=12

4y-x=8

Correct answer:

4y-x=8

Explanation:

First, put the equation of the given line in the y=mx+b form to find its slope.

3y=-12x+9

y=-4y+3

Since the slope of the given line is , the slope of the line that is perpendicular must be its negative reciprocal, $\frac{1}{4}$ .

Now, put each answer choice in y=mx+b form to see which one has a slope of $\frac{1}{4}$ .

4y-x=8

4y=x+8

$y=\frac{1}{4}x+2$

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Example Question #1 : How To Find Out If Lines Are Perpendicular

Which of the following lines is perpendicular to the line y+3x=7 ?

Possible Answers:

3y+9x=2

12y-4x=-6

6y+2x=14

2y-6x=4

$\frac{y}{3}-x=3$

Correct answer:

12y-4x=-6

Explanation:

Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.

y+3x=7

y=-3x+7

The slope of this line is . The negative reciprocal will be $+\frac{1}{3}$ , which will be the slope of the perpendicular line.

Now we need to find the answer choice with this slope by converting to slope-intercept form.

12y-4x=-6

12y=4x-6

$y=\frac{4}{12}x-\frac{6}{12}$

$y=\frac{1}{3}x-\frac{1}{2}$

This equation has a slope of $+\frac{1}{3}$ , and must be our answer.

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Example Question #2 : How To Find Out If Lines Are Perpendicular

Which of the following lines is perpindicular to y-3x=7

Possible Answers:

None of the other answers

$y=\frac{-1}{3}x - 4$

$y = \frac{1}{3}x +7$

y=-3x-2

$y-\frac{1}{3}x=12$

Correct answer:

$y=\frac{-1}{3}x - 4$

Explanation:

When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, y=mx+b , where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.

The first step of this problem is to get it into the form, y=mx+b , which is y=3x+7 . Now we know that the slope, m, is . The reciprocal of that is $\frac{1}{3}$ , and the negative of that is $\frac{-1}{3}$ . Therefore, any line that has a slope of $\frac{-1}{3}$ will be perpindicular to the original line.

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Example Question #2 : How To Find Out If Lines Are Perpendicular

Which of the following equations represents a line that is perpendicular to the line with points (5,-13) and (7,11) ?

Possible Answers:

y=-x+4

$y=\frac{1}{12}x+1$

y=-12x+6

$y=-\frac{1}{12}x+4$

y=12x-4

Correct answer:

$y=-\frac{1}{12}x+4$

Explanation:

If lines are perpendicular, then their slopes will be negative reciprocals.

First, we need to find the slope of the given line.

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{11-(-13)}{7-5}$

$m=\frac{24}{2}$

m=12

Because we know that our given line's slope is , the slope of the line perpendicular to it must be $-\frac{1}{12}$ .

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SSAT Upper Level Math : Coordinate Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

All SSAT Upper Level Math Resources

Example Questions

Example Question #2 : How To Find Out If A Point Is On A Line With An Equation

Example Question #3 : Other Lines

Example Question #52 : Lines

Example Question #51 : Coordinate Geometry

Example Question #52 : Coordinate Geometry

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #111 : Geometry

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #2 : How To Find Out If Lines Are Perpendicular

Example Question #2 : How To Find Out If Lines Are Perpendicular

All SSAT Upper Level Math Resources

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