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SAT Math : Coordinate Geometry

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Example Questions

Example Question #481 : Geometry

At what point do these two lines intersect?

2x-3y=1

3x+2y=2

Possible Answers:

(2, 1)

$(-\frac{8}{15}, \frac{3}{15})$

None of the given answers

$(\frac{8}{15}, \frac{3}{15})$

(1, 2)

Correct answer:

$(\frac{8}{15}, \frac{3}{15})$

Explanation:

If two lines intersect, that means that their and values are the same at one point. Therefore, we can use substitution to solve this problem.

First, let's write these two formulas in slope-intercept form. First:

2x-3y=1

-3y=-2x+1

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

Then, for the second line:

2y=-3x+2

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

Now, we can substitute $y=\frac{2}{3}x - \frac{1}{3}$ in for in our second equation and solve for , like so:

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

$\frac{2}{3}x +\frac{3}{2}x = \frac{4}{3}$

$\frac{4}{6}x +\frac{9}{6}x = \frac{4}{3}$

$\frac{15}{6}x = \frac{4}{3}$

$x = \frac{24}{45} = \frac{8}{15}$

Now, we can substitute this value into either equation to solve for .

2x - 3y = 1

$2(\frac{8}{15}) - 3y =1$

$\frac{16}{15} - 3y =1$

$-3y = 1-\frac{18}{15}$

$y = -3(\frac{15}{15} - \frac{16}{15}) = -3(-\frac{1}{15}) = \frac{3}{15}$

Therefore, our point of intersection is $(\frac{8}{15}, \frac{3}{15})$

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Example Question #102 : Coordinate Geometry

Trans

Lines P and Q are parallel. Find the value of .

Possible Answers:

$x=-5+ \sqrt{ 205}$

$x=-5- \sqrt{ 205}$

$x= \sqrt{ 205}$

$x=-5\pm \sqrt{ 205}$

$x=\pm \sqrt{ 205}$

Correct answer:

$x=-5\pm \sqrt{ 205}$

Explanation:

Since these are complementary angles, we can set up the following equation.

x^2+4+10x-4=180

x^2+10x=180

x^2+10x-180=0

Now we will use the quadratic formula to solve for .

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$x=\frac{-10\pm \sqrt{10^2-4(1)(-180)}}{2(1)}$

$x=\frac{-10\pm \sqrt{100+720}}{2}$

$x=\frac{-10\pm \sqrt{820}}{2}$

$x=\frac{-10\pm \sqrt{4\cdot 205}}{2}$

$x=\frac{-10\pm 2\sqrt{ 205}}{2}$

$x=-5\pm \sqrt{ 205}$

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Example Question #101 : Coordinate Geometry

$\begin{tabular}{||c c||} \hline Time & Distance \\ [0.5ex] \hline\hline 0&10 \\ 1&4 \\ 2&3 \\ 2.5 & 1 \\ 3&3\\ 4&4 \\ 5&10\\ \hline \hline \end{tabular}$

The table and graph describe two different particle's travel over time. Which particle has a lower minimum?

Possible Answers:

$\text{More information is needed}$

$\text{Graph}$

$\text{The table and graph are equal}$

$\text{Table}$

Correct answer:

$\text{Graph}$

Explanation:

This question is testing one's ability to compare the properties of functions when they are illustrated in different forms. This question specifically is asking for the examination and interpretation of two quadratic functions for which one is illustrated in a table format and the other is illustrated graphically.

Step 1: Identify the minimum of the table.

Using the table find the time value where the lowest distance exists.

$\begin{tabular}{||c c||} \hline Time & Distance \\ [0.5ex] \hline\hline 0&10 \\ 1&4 \\ 2&3 \\ {\color{Red} 2.5} & {\color{Red} 1} \\ 3&3\\ 4&4 \\ 5&10\\ \hline \hline \end{tabular}$

Recall that the time represents the values while the distance represents the values. Therefore the ordered pair for the minimum can be written as (2.5,1) .

Step 2: Identify the minimum of the graph

Recall that the minimum of a cubic function is known as a local minimum. This occurs at the valley where the vertex lies.

For this particular graph the vertex is at (1,-3) .

Step 3: Compare the minimums from step 1 and step 2.

Compare the value coordinate from both minimums.

$-3<1\Rightarrow \textup{Graph}< \textup{Table}$

Therefore, the graph has the lowest minimum.

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Example Question #11 : How To Find Out If A Point Is On A Line With An Equation

Axes

Figure NOT drawn to scale.

On the coordinate axes shown above, the shaded triangle has the following area:

Evaluate .

Possible Answers:

X = 4

$X = 3 \frac{3}{4}$

$X = 3 \frac{1}{4}$

$X = 3 \frac{1}{2}$

X = 3

Correct answer:

$X = 3 \frac{3}{4}$

Explanation:

The lengths of the horizontal and vertical legs of the triangle correspond to the -coordinate of the -intercept and the -coordinate of the -intercept. The area of a right triangle is half the product of the lengths of its legs and . The length of the vertical leg is b = 16 , so, setting b = 16 and A = 48 , and solving for :

$\frac{1}{2} ab = A$

$\frac{1}{2} \cdot a \cdot 16 = 48$

8a = 48

$\frac{8a }{8}= \frac{48}{8}$

a= 6

Therefore, the -intercept of the line containing the hypotenuse is (6,0 ) . The slope of the line given the coordinates of its intercepts is

$m = - \frac{b}{a}$ .

substituting:

$m = - \frac{16}{6} = -\frac{8}{3}$ .

Substituting for and in the slope-intercept form of the equation of a line,

y = mx+ b ,

the line has equation

$y = -\frac{8}{3} x+16$ .

Substituting for and 6 for and solving for , we find the -coordinate

of the point on the line with -coordinate 6:

$6 = -\frac{8}{3} X+16$

$6 + \frac{8}{3} X - 6 = -\frac{8}{3} X+16 + \frac{8}{3} X - 6$

$\frac{8}{3} X = 10$

$\frac{3} {8}\cdot \frac{8}{3} X = \frac{3} {8}\cdot 10$

$X = \frac{15} {4} = 3 \frac{3}{4}$

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Example Question #1 : Coordinate Geometry

Based on the table below, when x = 5, y will equal

x	y
-1	3
0	1
1	-1
2	-3

Possible Answers:

–11

–9

–10

Correct answer:

–9

Explanation:

Use 2 points from the chart to find the equation of the line.

Example: (–1, 3) and (1, –1)

Using the formula for the slope, we find the slope to be –2. Putting that into our equation for a line we get y = –2x + b. Plug in one of the points for x and y into this equation in order to find b. b = 1.

The equation then will be: y = –2x + 1.

Plug in 5 for x in order to find y.

y = –2(5) + 1

y = –9

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Example Question #1 : Coordinate Geometry

What is the slope of a line that runs through points: (-2, 5) and (1, 7)?

Possible Answers:

2/3

5/7

3/2

Correct answer:

2/3

Explanation:

The slope of a line is defined as a change in the y coordinates over a change in the x coordinates (rise over run).

To calculate the slope of a line, use the following formula:

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Example Question #2 : Coordinate Geometry

A line passes through the points (–3, 5) and (2, 3). What is the slope of this line?

Possible Answers:

2/3

–2/5

–2/3

2/5

-3/5

Correct answer:

–2/5

Explanation:

The slope of the line that passes these two points are simply ∆y/∆x = (3-5)/(2+3) = -2/5

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Example Question #2 : Geometry

Which of the following lines intersects the y-axis at a thirty degree angle?

Possible Answers:

y = x - √2

y = x

y = x√3 + 2

y = x√2 - 2

y = x((√3)/3) + 1

Correct answer:

y = x√3 + 2

Explanation:

Line_intersect1

Line_intersect2

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Example Question #5 : How To Find Slope Of A Line

What is a possible slope of line y?

Possible Answers:

$\dpi{100} \small -\frac{1}{2}$

$\dpi{100} \small \frac{1}{2}$

–2

Correct answer:

–2

Explanation:

The slope is negative as it starts in quadrant 2 and ends in quadrant 4. Slope is equivlent to the change in y divided by the change in x. The change in y is greater than the change in x, which implies that the slope must be less than –1, leaving –2 as the only possible solution.

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Example Question #1 : Slope And Line Equations

What is the slope between (8,3) and (5,7) ?

Possible Answers:

$-\frac{4}{3}$

$-\frac{5}{2}$

$\frac{3}{4}$

$\frac{2}{5}$

$\frac{1}{6}$

Correct answer:

$-\frac{4}{3}$

Explanation:

Let $P_{1}=(8,3)$ and $P_{2}=(5,7)$

$m = (y_{2} - y_{1}) \div (x_{2} - x_{1})$ so the slope becomes $-\frac{4}{3}$ .

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SAT Math : Coordinate Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #481 : Geometry

Example Question #102 : Coordinate Geometry

Example Question #101 : Coordinate Geometry

Example Question #11 : How To Find Out If A Point Is On A Line With An Equation

Example Question #1 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

Example Question #2 : Coordinate Geometry

Example Question #2 : Geometry

Example Question #5 : How To Find Slope Of A Line

Example Question #1 : Slope And Line Equations

All SAT Math Resources

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