Midpoint Formula - GMAT Quantitative

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Question

Consider segment with midpoint at the point .

I) Point has coordinates of .

II) Segment has a length of units.

What are the coordinates of point ?

Answer

In this case, we are given the midpoint of a line and asked to find one endpoint.

Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.

Midpoint formula: $\small (x',y')=\left(\frac{x^1-x^2}{2}, \frac{y^1-y^2}{2}\right)$

Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.

So,

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

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Question

Find endpoint given the following:

I) Segment has its midpoint at .

II) Point is located on the -axis, points from the origin.

Answer

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

$(45,65)=\left(\frac{x_1-13}{2},\frac{y_1-0}{2}\right)$

$45\cdot2=x_1-13$

$65\cdot2=y_1=130$

So endpoint is at .

Therefore, both statements are needed to answer the question.

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Question

Consider segment

I) Endpoint is located at the origin

II) has a distance of 36 units

Where is endpoint located?

Answer

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of without more information.

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Question

is the midpoint of line PQ. What are the coordinates of point P?

(1) Point Q is the origin.

(2) Line PQ is 8 units long.

Answer

The midpoint formula is

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

with statement 1, we know that Q is $\left ( 0,0\right )$ and can solve for P:

$\frac{0+x}{}2=2$ and $\frac{0+y}{}2=3$

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

$p=\left ( 4,6\right )$

Statement 1 alone is sufficient.

Statement 2 doesn't provide enough information to solve for point P.

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Question

In what quadrant or axis is the midpoint of the line segment with endpoints and located?

Statement 1:

Statement 2: is in Quadrant IV.

Answer

The midpoint of the segment with endpoints and is $\left ( \frac{A+B}{2}, \frac{A+B}{2} \right )$ .

If , then and $\frac{A+B}{2}>0$ , so the midpoint, having both of its coordinates positive, is in Quadrant I.

If is in Quadrant IV, then and . But the quadrant of the midpoint varies according to and :

Example 1: If , the midpoint is $\left ( \frac{8+(-4) }{2}, \frac{8+(-4) }{2} \right )$ , or , putting it in Quadrant I.

Example 2: If , the midpoint is $\left ( \frac{4-8 }{2}, \frac{4-8 }{2} \right )$ , or , putting it in Quadrant III.

Therefore, the first statement, but not the second, tells us all we need to know.

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Question

A line segment has one of its endpoints at . In which quadrant, or on what axis, is its other endpoint?

Statement 1: The midpoint of the segment is .

Statement 2: The length of the segment is 10.

Answer

Statement 1 give us the means to find the other endpoint using the midpoint formula:

$x _{m } =\frac{x _{1 } + x _{2 }}{2}$

$9 =\frac{13 + x _{2 }}{2}$

$18 =12 + x _{2 }$

$x _{2 } = 6$

Similarly,

$y _{m } =\frac{y _{1 } + y _{2 }}{2}$

$9 =\frac{12 + y _{2 }}{2}$

$18 =13 + y _{2 }$

$y_{2 } = 5$

This makes the endpoint , which is in Quadrant I.

Statement 2 is also sufficient. , which is in Quadrant 1, is 12 units away from the nearest axis; since the length of the segment is 10, the entire segment must be in Quadrant I.

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Question

Consider segment . What are the coordinates of the midpoint of ?

I) Point has coordinates of .

II) Point has coordinates of .

Answer

We are asked to find the midpoint of a line segment and given endpoints in our clues.

Midpoint formula is found by taking the average of the x and y values of two points.

$midpoint=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right )$

We need both endpoints to solve this problem, so both statements are needed.

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Question

Find the midpoint of segment given that point is at .

I) The coordinate of is twice that of , and the coordinate of is $\frac{3}{4}^{th}$ that of .

II) is $2\sqrt{5}$ units long.

Answer

To find the midpoint, we need to know both endpoints.

I) Gives us the means to find out other endpoint.

II) Gives us the length of PS, but we are not given any hint as to its orientation.Thus, we cannot find the other endpoint and we cannot find the midpoint.

Thus, Statement I alone is sufficient to answer the question.

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Question

Find the midpoint of segment .

I) Endpoint has coordinates of .

II) Endpoint coordinate is half of , and coordinate is one sixteenth of coordinate.

Answer

To find the midpoint of a segment we need both endpoints

I) Gives us one endpoint.

II) Gives us clues to find the other endpoint.

has coordinates of

Use midpoint formula

$(x_m,y_m)=\left(\frac{x_1-x_2}{2},\frac{y_1-y_2}{2}\right)$

$(x_m,y_m)=\left(\frac{76-38}{2},\frac{148-9.25}{2}\right)=(19,69.375)$

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Question

In what quadrant or axis is the midpoint of the line segment with endpoints and located?

Statement 1:

Statement 2: is in Quadrant IV.

Answer

The midpoint of the segment with endpoints and is $\left ( \frac{A+B}{2}, \frac{A+B}{2} \right )$ .

If , then and $\frac{A+B}{2}>0$ , so the midpoint, having both of its coordinates positive, is in Quadrant I.

If is in Quadrant IV, then and . But the quadrant of the midpoint varies according to and :

Example 1: If , the midpoint is $\left ( \frac{8+(-4) }{2}, \frac{8+(-4) }{2} \right )$ , or , putting it in Quadrant I.

Example 2: If , the midpoint is $\left ( \frac{4-8 }{2}, \frac{4-8 }{2} \right )$ , or , putting it in Quadrant III.

Therefore, the first statement, but not the second, tells us all we need to know.

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Question

A line segment has one of its endpoints at . In which quadrant, or on what axis, is its other endpoint?

Statement 1: The midpoint of the segment is .

Statement 2: The length of the segment is 10.

Answer

Statement 1 give us the means to find the other endpoint using the midpoint formula:

$x _{m } =\frac{x _{1 } + x _{2 }}{2}$

$9 =\frac{13 + x _{2 }}{2}$

$18 =12 + x _{2 }$

$x _{2 } = 6$

Similarly,

$y _{m } =\frac{y _{1 } + y _{2 }}{2}$

$9 =\frac{12 + y _{2 }}{2}$

$18 =13 + y _{2 }$

$y_{2 } = 5$

This makes the endpoint , which is in Quadrant I.

Statement 2 is also sufficient. , which is in Quadrant 1, is 12 units away from the nearest axis; since the length of the segment is 10, the entire segment must be in Quadrant I.

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Question

Consider segment . What are the coordinates of the midpoint of ?

I) Point has coordinates of .

II) Point has coordinates of .

Answer

We are asked to find the midpoint of a line segment and given endpoints in our clues.

Midpoint formula is found by taking the average of the x and y values of two points.

$midpoint=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right )$

We need both endpoints to solve this problem, so both statements are needed.

Compare your answer with the correct one above

Question

Find the midpoint of segment given that point is at .

I) The coordinate of is twice that of , and the coordinate of is $\frac{3}{4}^{th}$ that of .

II) is $2\sqrt{5}$ units long.

Answer

To find the midpoint, we need to know both endpoints.

I) Gives us the means to find out other endpoint.

II) Gives us the length of PS, but we are not given any hint as to its orientation.Thus, we cannot find the other endpoint and we cannot find the midpoint.

Thus, Statement I alone is sufficient to answer the question.

Compare your answer with the correct one above

Question

Find the midpoint of segment .

I) Endpoint has coordinates of .

II) Endpoint coordinate is half of , and coordinate is one sixteenth of coordinate.

Answer

To find the midpoint of a segment we need both endpoints

I) Gives us one endpoint.

II) Gives us clues to find the other endpoint.

has coordinates of

Use midpoint formula

$(x_m,y_m)=\left(\frac{x_1-x_2}{2},\frac{y_1-y_2}{2}\right)$

$(x_m,y_m)=\left(\frac{76-38}{2},\frac{148-9.25}{2}\right)=(19,69.375)$

Compare your answer with the correct one above

Question

Consider segment with midpoint at the point .

I) Point has coordinates of .

II) Segment has a length of units.

What are the coordinates of point ?

Answer

In this case, we are given the midpoint of a line and asked to find one endpoint.

Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.

Midpoint formula: $\small (x',y')=\left(\frac{x^1-x^2}{2}, \frac{y^1-y^2}{2}\right)$

Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.

So,

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Compare your answer with the correct one above

Question

Find endpoint given the following:

I) Segment has its midpoint at .

II) Point is located on the -axis, points from the origin.

Answer

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

$(45,65)=\left(\frac{x_1-13}{2},\frac{y_1-0}{2}\right)$

$45\cdot2=x_1-13$

$65\cdot2=y_1=130$

So endpoint is at .

Therefore, both statements are needed to answer the question.

Compare your answer with the correct one above

Question

Consider segment

I) Endpoint is located at the origin

II) has a distance of 36 units

Where is endpoint located?

Answer

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of without more information.

Compare your answer with the correct one above

Question

is the midpoint of line PQ. What are the coordinates of point P?

(1) Point Q is the origin.

(2) Line PQ is 8 units long.

Answer

The midpoint formula is

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

with statement 1, we know that Q is $\left ( 0,0\right )$ and can solve for P:

$\frac{0+x}{}2=2$ and $\frac{0+y}{}2=3$

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

$p=\left ( 4,6\right )$

Statement 1 alone is sufficient.

Statement 2 doesn't provide enough information to solve for point P.

Compare your answer with the correct one above

Question

Consider segment with midpoint at the point .

I) Point has coordinates of .

II) Segment has a length of units.

What are the coordinates of point ?

Answer

In this case, we are given the midpoint of a line and asked to find one endpoint.

Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.

Midpoint formula: $\small (x',y')=\left(\frac{x^1-x^2}{2}, \frac{y^1-y^2}{2}\right)$

Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.

So,

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Compare your answer with the correct one above

Question

Find endpoint given the following:

I) Segment has its midpoint at .

II) Point is located on the -axis, points from the origin.

Answer

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

$(45,65)=\left(\frac{x_1-13}{2},\frac{y_1-0}{2}\right)$

$45\cdot2=x_1-13$

$65\cdot2=y_1=130$

So endpoint is at .

Therefore, both statements are needed to answer the question.

Compare your answer with the correct one above

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