DSQ: Calculating the midpoint of a line segment - GMAT Quantitative

Card 0 of 35

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

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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

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.

Compare your answer with the correct one above

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.

Compare your answer with the correct one above

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

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