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SAT Math : Coordinate Geometry

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Example Questions

Example Question #651 : Geometry

On the coordinate plane, a triangle has its vertices at the points with coordinates (0,0) , (12, 0) , and (0,8) . Give the coordinates of the center of the circle that circumscribes this triangle.

Possible Answers:

(6, 4)

(12, 8)

$\left ( 4, 2 \frac{2}{3}\right )$

$\left ( 8, 5 \frac{1}{3}\right )$

(0,0)

Correct answer:

(6, 4)

Explanation:

The referenced figure is below.

Triangle a

The triangle formed is a right triangle whose hypotenuse is the segment with the endpoints (0, 8) and (12,0) . The center of the circle that circumscribes a right triangle is the midpoint of its hypotenuse, so the midpoint formula

$(x_{M}, y_{M}) = \left ( \frac{x_{1}+ x_{2} }{2},\frac{y_{1}+ y_{2} }{2} \right )$

can be applied, setting $x_{1} = 12, y_{1} = x_{2} = 0, y_{2} = 8$ :

$x_{M}= \frac{x_{1}+ x_{2} }{2}= \frac{12+0}{2}= \frac{12}{2} = 6$

$y_{M}= \frac{y_{1}+ y_{2} }{2}= \frac{0+8}{2}= \frac{8}{2} = 4$

The midpoint of the hypotenuse, and, consequently, the center of the circumscribed circle, is the point with coordinates (6, 4) .

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Example Question #271 : Coordinate Geometry

On the coordinate plane, , , and are the points with coordinates (0, 0) , (0, 12) , and (6, 6) , respectively. Lines , , and are the perpendicular bisectors of $\overline{XY}$ , $\overline{XZ}$ , and $\overline{YZ}$ , respectively.

and intersect at a point ; and intersect at a point ; and intersect at a point .

Which of these statements is true of , , and ?

Possible Answers:

and are the same point; is a different point.

, , and are distinct and collinear.

, , and are distinct and are the vertices of a triangle similar to $\bigtriangleup XYZ$ .

, , and are the same point.

, , and are distinct and are the vertices of an equilateral triangle.

Correct answer:

, , and are the same point.

Explanation:

Another way of viewing this problem is to note that the three given vertices form a triangle $\bigtriangleup XYZ$ whose sides' perpendicular bisectors intersect at the points , , and . However, the three perpendicular bisectors of the sides of any triangle always intersect at a common point. The correct response is that , , and are the same point.

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Example Question #655 : Geometry

Axes

Figure NOT drawn to scale.

On the coordinate axes shown above, the shaded triangle has area 40.

Evaluate .

Possible Answers:

$Y = 8 \frac{3}{4}$

Y = 9

Y = 6

Y = 8

$Y= 7 \frac{1}{2}$

Correct answer:

$Y= 7 \frac{1}{2}$

Explanation:

The length of the horizontal leg of the triangle is the distance from the origin (0,0) to (8,0) , which is 8.

The area of a right triangle is half the product of the lengths of its legs and , so, setting a = 8 and A = 40 and solving for :

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

$\frac{1}{2} \cdot 8 \cdot b = 40$

4 b = 40

$\frac{4 b}{4} = \frac{40}{4}$

b = 10

Therefore, the length of the vertical leg is 10, and, since the -intercept of the line containing the hypotenuse is on the positive -axis, this intercept is (0, 10 ) . The slope of a line with intercepts (a, 0),(0, b) is

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

so, setting a = 8 and b = 10 :

$m = - \frac{10}{8} = -\frac{5}{4}$

Set $m = -\frac{5}{4}$ and b = 10 in the slope-intercept form of the equation of a line,

y = mx+ b ;

the line has equation

$y = -\frac{5}{4}x+10$

The -coordinate of the point on the line with -coordinate 2 can be found using substitution; setting x= 2 , y = Y ::

$Y = -\frac{5}{4} (2)+10 = -\frac{5}{2} + 10 = -2\frac{1}{2} + 10 = 7 \frac{1}{2}$

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Example Question #661 : Sat Mathematics

Mrs. Smith's 8th grade class has a weekly quiz. The graph below depicts the number of questions students got incorrect on their quiz and their corresponding quiz grade. Examining the graph, what type of correlation if any, exists?

Screen shot 2016 02 10 at 10.06.45 am

Possible Answers:

The graph depicts no correlation.

The graph depicts a positive correlation.

The graph depicts a negative correlation.

The graph depicts a constant correlation.

More information is needed.

Correct answer:

The graph depicts a negative correlation.

Explanation:

Mrs. Smith's 8th grade class had a quiz last week. The graph below depicts the number of questions students got incorrect on their quiz and their corresponding quiz grade. In other words, the graph in this particular question is a dot plot and the question asks to find a correlation if one exists.

Recall that a correlation is a trend seen in the data. Graphically, trends can be either:

I. Positive

II. Negative

III. Constant

IV. No trend

For a trend to be positive the x and y variable both increase. A trend is negative when the y variable (dependent variable) decreases as the x variable (independent variable) increases. A constant trend occurs when the y variable stays the same as the x variable increases. No trend exists when the data appears to be scattered with no association between the x and y variables.

Screen shot 2016 02 10 at 10.06.45 am

Examining the graph given it is seen that the x variable is the number of questions missed and the y variable is the overall score on the quiz. It is seen that as the number of questions missed increases, the overall score on the quiz decreases. This describes a negative trend.

In other words, the graph depicts a negative correlation.

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Example Question #12 : Graphing

Given the graph of record sales, what fraction of records were sold in 2004 to 2010?

Screen shot 2016 02 11 at 6.41.34 am

Possible Answers:

$\frac{12}{13}$

$\frac{13}{14}$

$\frac{9}{10}$

$\frac{14}{13}$

$\frac{11}{14}$

Correct answer:

$\frac{14}{13}$

Explanation:

Given the graph of record sales, to find the fraction of records that were sold in 2004 to 2010 first identify the record sales in 2004 and the record sales in 2010. Screen shot 2016 02 11 at 6.41.34 am

Examining the graph,

Record sales in 2004:14 million

Record sales in 2010: 13 million

From here, to find the fraction of records sold during this time period, use the following formula.

$\text{Fraction of Sales}=\frac{\text{Beginning Year Record Sales}}{\text{Ending Year Record Sales}}$

$\text{Fraction}=\frac{14}{13}$

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Example Question #1 : How To Graph An Ordered Pair

Which of the following coordinate pairs is farthest from the origin?

Possible Answers:

(-1, 8)

(0, 7)

(2, 5)

(1, 6)

(3, 4)

Correct answer:

(-1, 8)

Explanation:

Using the distance formula, calculate the distance from each of these points to the origin, (0, 0). While each answer choice has coordinates that add up to seven, (-1, 8) is the coordinate pair that produces the largest distance, namely $\sqrt{65}$ , or approximately 8.06.

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Example Question #11 : Graphing

On the coordinate plane, the point with coordinates $\left ( -5 \frac{1}{4},\ 6\frac{4}{7}\right )$ is located in __________.

Possible Answers:

Quadrant I

None of these

Quadrant III

Quadrant IV

Quadrant II

Correct answer:

Quadrant II

Explanation:

On the coordinate plane, a point with a negative -coordinate and a positive -coordinate lies in the upper left quadrant—Quadrant II.

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Example Question #111 : Geometry

A line graphed on the coordinate plane below. Graph_of_y_-2x_4

Give the equation of the line in slope intercept form.

Possible Answers:

$\dpi{100} \small y=2x+4$

$\dpi{100} \small y=-2x+4$

$\dpi{100} \small y=-x+4$

$\dpi{100} \small y=-2x-4$

$\dpi{100} \small y=2x-4$

Correct answer:

$\dpi{100} \small y=-2x+4$

Explanation:

The slope of the line is $\dpi{100} \small -2$ and the y-intercept is $\dpi{100} \small 4$ .

The equation of the line is $\dpi{100} \small y=-2x+4$ .

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Example Question #2 : Graphing

Graph_of_y_-x_3

Give the equation of the curve.

Possible Answers:

$\dpi{100} \small y=-x^{2}$

$\dpi{100} \small y=x^{4}$

$\dpi{100} \small y=x^{3}$

$\dpi{100} \small y=-x^{3}$

None of the other answers

Correct answer:

$\dpi{100} \small y=-x^{3}$

Explanation:

Graph_of_x_3 This is the parent graph of $\dpi{100} \small x^{3}$ . Since the graph in question is negative, then we flip the quadrants in which it will approach infinity. So the graph of $\dpi{100} \small y=-x^{3}$ will start in quadrant 2 and end in 4.

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Example Question #16 : Graphing

Give the area of the triangle on the coordinate plane that is bounded by the lines of the equations y = -2 , y = x and y = 4x- 7 .

Possible Answers:

$7\frac{7}{12}$

$3\frac{19}{24}$

$1 \frac{11}{24}$

$7 \frac{1}{24}$

$2 \frac{11}{12}$

Correct answer:

$7 \frac{1}{24}$

Explanation:

It is necessary to find the coordinates of the vertices of the triangle, each of which is the intersection of two of the three lines.

The intersection of the lines of the equations y = -2 and y = x can be found by noting that, by substituting for in the latter equation, -2 = x , making the point of intersection (-2,-2) .

The intersection of the lines of the equations y = -2 and y = 4x- 7 can be found by substituting for in the latter equation and solving for :

4x- 7 = -2

4x- 7 + 7 = -2 + 7

4x = 5

$\frac{4x}{4} = \frac{5}{4}$

$x = 1 \frac{1}{4}$

This point of intersection is $\left ( 1 \frac{1}{4}, -2 \right )$ .

The intersection of the lines of the equations y = x and y = 4x- 7 can be found by substituting for in the latter equation and solving for :

4x- 7 = x

4x- 7 - x + 7 = x - x + 7

3x = 7

$\frac{3x }{3}= \frac{7}{3}$

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

Since y = x , $y = 2\frac{1}{3}$ , and this point of intersection is $\left ( 2\frac{1}{3}, 2\frac{1}{3} \right )$ .

The lines in question are graphed below, and the triangle they bound is shaded:

Triangle z

We can take the horizontal side as the base of the triangle; its length is the difference of the -coordinates:

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

The height is the vertical distance from this side to the opposite side, which is the difference of the -coordinates:

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

The area is half their product:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 3 \frac{1}{4} \cdot 4 \frac{1}{3} =7 \frac{1}{24}$

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SAT Math : Coordinate Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #651 : Geometry

Example Question #271 : Coordinate Geometry

Example Question #655 : Geometry

Example Question #661 : Sat Mathematics

Example Question #12 : Graphing

Example Question #1 : How To Graph An Ordered Pair

Example Question #11 : Graphing

Example Question #111 : Geometry

Example Question #2 : Graphing

Example Question #16 : Graphing

All SAT Math Resources

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