Coordinate Geometry - GRE Quantitative Reasoning

Card 0 of 936

Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{4+a}{2},\frac{3+b}{2}\right)=(7,15)$

You merely need to solve each coordinate for its respective value.

$\frac{4+a}{2}=7$

Then, for the y-coordinate:

$\frac{3+b}{2}=15$

Therefore, our other point is:

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Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{11+a}{2},\frac{-5+b}{2}\right)=(-6,-21)$

You merely need to solve each coordinate for its respective value.

$\frac{11+a}{2}=-6$

Then, for the y-coordinate:

$\frac{-5+b}{2}=-21$

Therefore, our other point is:

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Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{6+c}{2},\frac{8+d}{2}\right)=(14,1)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{6+c}{2} = 14$

Y-Coordinate:

$\frac{8+d}{2}=1$

Therefore, our point is

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Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{-15+c}{2},\frac{14+d}{2}\right)=(-19,4)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{-15+c}{2}=-19$

Y-Coordinate:

$\frac{14+d}{2}=4$

Therefore, our point is

Compare your answer with the correct one above

Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{4+a}{2},\frac{3+b}{2}\right)=(7,15)$

You merely need to solve each coordinate for its respective value.

$\frac{4+a}{2}=7$

Then, for the y-coordinate:

$\frac{3+b}{2}=15$

Therefore, our other point is:

Compare your answer with the correct one above

Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{11+a}{2},\frac{-5+b}{2}\right)=(-6,-21)$

You merely need to solve each coordinate for its respective value.

$\frac{11+a}{2}=-6$

Then, for the y-coordinate:

$\frac{-5+b}{2}=-21$

Therefore, our other point is:

Compare your answer with the correct one above

Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{6+c}{2},\frac{8+d}{2}\right)=(14,1)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{6+c}{2} = 14$

Y-Coordinate:

$\frac{8+d}{2}=1$

Therefore, our point is

Compare your answer with the correct one above

Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{-15+c}{2},\frac{14+d}{2}\right)=(-19,4)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{-15+c}{2}=-19$

Y-Coordinate:

$\frac{14+d}{2}=4$

Therefore, our point is

Compare your answer with the correct one above

Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{4+a}{2},\frac{3+b}{2}\right)=(7,15)$

You merely need to solve each coordinate for its respective value.

$\frac{4+a}{2}=7$

Then, for the y-coordinate:

$\frac{3+b}{2}=15$

Therefore, our other point is:

Compare your answer with the correct one above

Question

There is a line defined by two end-points, and . The midpoint between these two points is . What is the value of the point ?

Answer

Recall that to find the midpoint of two points and , you use the equation:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$ .

(It is just like finding the average of the two points, really.)

So, for our equation, we know the following:

$\left(\frac{11+a}{2},\frac{-5+b}{2}\right)=(-6,-21)$

You merely need to solve each coordinate for its respective value.

$\frac{11+a}{2}=-6$

Then, for the y-coordinate:

$\frac{-5+b}{2}=-21$

Therefore, our other point is:

Compare your answer with the correct one above

Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{6+c}{2},\frac{8+d}{2}\right)=(14,1)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{6+c}{2} = 14$

Y-Coordinate:

$\frac{8+d}{2}=1$

Therefore, our point is

Compare your answer with the correct one above

Question

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Answer

What is the other endpoint of a line segment with one point that is and a midpoint of ?

Recall that the midpoint formula is like finding the average of the and values for two points. For two points and , it is:

$\left(\frac{a+c}{2},\frac{b+d}{2}\right)$

For our points, we are looking for . We know:

$\left(\frac{-15+c}{2},\frac{14+d}{2}\right)=(-19,4)$

We can solve for each of these coordinates separately:

X-Coordinate

$\frac{-15+c}{2}=-19$

Y-Coordinate:

$\frac{14+d}{2}=4$

Therefore, our point is

Compare your answer with the correct one above

Question

What is the slope of the line whose equation is $\dpi{100} \small 8x+12y=20$ ?

Answer

Solve for $\dpi{100} \small y$ so that the equation resembles the $\dpi{100} \small y=mx+b$ form. This equation becomes $\dpi{100} \small -\frac{2}{3}x+\frac{5}{3}$ . In this form, the $\dpi{100} \small m$ is the slope, which is $\dpi{100} \small -\frac{2}{3}$ .

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Question

Which of the following equations has a -intercept of ?

Answer

To find the -intercept, you need to find the value of the equation where . The easiest way to do this is to substitute in for your value of and see where you get for . If you do this for each of your equations proposed as potential answers, you find that is the answer.

Substitute in for :

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Question

If is a line that has a -intercept of and an -intercept of , which of the following is the equation of a line that is perpendicular to ?

Answer

If has a -intercept of , then it must pass through the point .

If its -intercept is , then it must through the point .

The slope of this line is $\frac{0-3}{7-0}=-\frac{3}{7}$ .

Therefore, any line perpendicular to this line must have a slope equal to the negative reciprocal, which is $\frac{7}{3}$ . Only $y=\frac{(7x+15)}{3}$ has a slope of $\frac{7}{3}$ .

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Question

What is the slope of the line whose equation is $\dpi{100} \small 8x+12y=20$ ?

Answer

Solve for $\dpi{100} \small y$ so that the equation resembles the $\dpi{100} \small y=mx+b$ form. This equation becomes $\dpi{100} \small -\frac{2}{3}x+\frac{5}{3}$ . In this form, the $\dpi{100} \small m$ is the slope, which is $\dpi{100} \small -\frac{2}{3}$ .

Compare your answer with the correct one above

Question

Which of the following equations has a -intercept of ?

Answer

To find the -intercept, you need to find the value of the equation where . The easiest way to do this is to substitute in for your value of and see where you get for . If you do this for each of your equations proposed as potential answers, you find that is the answer.

Substitute in for :

Compare your answer with the correct one above

Question

If is a line that has a -intercept of and an -intercept of , which of the following is the equation of a line that is perpendicular to ?

Answer

If has a -intercept of , then it must pass through the point .

If its -intercept is , then it must through the point .

The slope of this line is $\frac{0-3}{7-0}=-\frac{3}{7}$ .

Therefore, any line perpendicular to this line must have a slope equal to the negative reciprocal, which is $\frac{7}{3}$ . Only $y=\frac{(7x+15)}{3}$ has a slope of $\frac{7}{3}$ .

Compare your answer with the correct one above

Question

What is the slope of the line whose equation is $\dpi{100} \small 8x+12y=20$ ?

Answer

Solve for $\dpi{100} \small y$ so that the equation resembles the $\dpi{100} \small y=mx+b$ form. This equation becomes $\dpi{100} \small -\frac{2}{3}x+\frac{5}{3}$ . In this form, the $\dpi{100} \small m$ is the slope, which is $\dpi{100} \small -\frac{2}{3}$ .

Compare your answer with the correct one above

Question

Which of the following equations has a -intercept of ?

Answer

To find the -intercept, you need to find the value of the equation where . The easiest way to do this is to substitute in for your value of and see where you get for . If you do this for each of your equations proposed as potential answers, you find that is the answer.

Substitute in for :

Compare your answer with the correct one above

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