Midpoint Formula - ACT Math

Card 0 of 19

Question

In the standard (x,y) coordinate plane, the midpoint of line XY is (12, –3) and point X is located at (3, 4). What are the coordinates of point Y?

Answer

To get from the midpoint of (12, –3) to point (3,4), we travel –9 units in the x-direction and 7 units in the y-direction. To find the other point, we travel the same magnitude in the opposite direction from the midpoint, 9 units in the x-direction and –7 units in the y-direction to point (21, –10).

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Question

The midpoint of a line segment is . If one endpoint of the line segment is , what is the other endpoint?

Answer

The midpoint formula can be used to solve this problem, where the midpoint is the average of the two coordinates.

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

We are given the midpoint and one endpoint. Plug these values into the formula.

$(4,-1)=(\frac{-4+x}{2},\frac{3+y}{2})$

$4=\frac{-4+x}{2}\ \text{and}\ -1=\frac{3+y}{2}$

Solve for the variables to find the coordinates of the second endpoint.

$8=-4+x\ \text{and}\ -2=3+y$

$12=x\ \text{and}\ -5=y$

The final coordinates of the other endpoint are .

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Question

Suppose the midpoint of a line segment is What are the endpoints of the segment?

Answer

The midpoint of a line segment is found using the formula $\left ( \frac{x_1+x_2 }{2}, \frac{y_1+y_2}{2}\right )$ .

The midpoint is given as Going through the answer choices, only the points and yield the correct midpoint of .

$\left ( \frac{2+6}{2}, \frac{8+16}{2} \right )=(4,12)$

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Question

What is the midpoint of the segment of

between and ?

Answer

What is the midpoint of the segment of

between and ?

To find this midpoint, you must first calculate the two end points. Thus, substitute in for :

Then, for :

Thus, the two points in question are:

and

The midpoint of two points is:

$(\frac{x_1+x_0}{2},\frac{y_1+y_0}{2})$

Thus, for our data, this is:

$(\frac{4+10}{2},\frac{17+26}{2})$

or

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Question

If is the midpoint of and another point, what is that other point?

Answer

If is the midpoint of and another point, what is that other point?

Recall that the midpoint's and values are the average of the and values of the two points in question. Thus, if we call the other point , we know that:

$15 = \frac{5+x}{2}$ and $40 = \frac{10+y}{2}$

Solve each equation accordingly:

For $15 = \frac{5+x}{2}$ , multiply both sides by and then subtract from both sides:

Thus,

For $40 = \frac{10+y}{2}$ , multiply both sides by 2 and then subtract 10 from both sides:

Thus,

Thus, our point is

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Question

If is the midpoint of and another point, what is that other point?

Answer

Recall that the midpoint's and values are the average of the and values of the two points in question. Thus, if we call the other point , we know that:

$-10 = \frac{-20+x}{2}$ and $-3 = \frac{15+y}{2}$

Solve each equation accordingly:

For $-10 = \frac{-20+x}{2}$ , multiply both sides by :

Thus,

The same goes for the other equation:

, so

Thus, our point is

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Question

What is the midpoint of MN between the points M(2, 6) and N (8, 4)?

Answer

The midpoint formula is equal to . Add the x-values together and divide them by 2, and do the same for the y-values.

x: (2 + 8) / 2 = 10 / 2 = 5

y: (6 + 4) / 2 = 10 / 2 = 5

The midpoint of MN is (5,5).

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Question

In the standard coordinate plane, what is the midpoint of a line segment that goes from the point (3, 5) to the point (7, 9)?

Answer

The midpoint formula is . An easy way to remember this is that finding the midpoint simply requires that you find the averageof the two x-coordinates and the average of the two y-coordinates. In this case, the two x-coordinates are 3 and 7, and the two y-coordinates are 5 and 9. If we substitute these values into the midpoint formula, we get (3 + 7/2), (5 + 9)/2, which equals (5, 7). If you got (–2, –2), you may have subtracted your x and y-coordinates instead of adding. If you got (10,14), you may have forgotten to divide your x and y-coordinates by 2. If you got (6,6), you may have found the average of x1 and y2 and x2 and y1 instead of keeping the x-coordinates together and the y-coordinates together. If you got (7, 5), you may have switched the x and y-coordinates.

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Question

Janice and Mark work in a city with neatly gridded streets. If Janice works at the intersection of 33rd Street and 7th Avenue, and Mark works at 15th Street and 5th Avenue, how many blocks will they each travel to lunch if they meet at the intersection exactly in between both offices?

Answer

Translating the intersections into points on a graph, Janice works at (33,7) and Mark works at (15,5). The midpoint of these two points is found by taking the average of the x-coordinates and the average of the y-coordinates, giving ((33+15)/2 , (5+7)/2) or (24, 6). Traveling in one direction at a time, the number of blocks from either office to 24th street is 9, and the number of blocks to 6th is 1, for a total of 10 blocks.

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Question

What is the coordinate of the point that is halfway between (-2, -4) and (6, 4)?

Answer

The midpoint formula is

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Question

What is the midpoint between and ?

Answer

Using the midpoint formula, $m=(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$

We get: $(\frac{-4+3}{2},\frac{-6+7}{2})$

Which becomes: which becomes $(-\frac{1}{2},\frac{1}{2})$

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Question

Find the midpoint of a line segment with endpoints (–1, 4) and (3, 6).

Answer

The formula for midpoint = (x1 + x2)/2, (y1 + y2)/2. Substituting in the two x coordinates and two y coordinates from the endpoints, we get (–1 + 3)/2.

(4 + 6)/2 or (1, 5) as the midpoint.

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Question

In the standard x, y coordinate plane what are the coordinates of the midpoint of a line whose endpoints are (–6, 4) and (4, –6)?

Answer

To solve this problem we use the midpoint formula. We find the average of the x and y coordinates. (–6 + 4)/2, (4 + –6)/2 = –1, –1

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Question

On the real number line, what is the midpoint between $-7$ and $19$ ?

Answer

On the number line, $-7$ is $26$ units away from $19$ .

We find the midpoint of this distance by dividing it by 2.

$\frac{26}{2}=13$

To find the midpoint, we add this value to the smaller number or subtract it from the larger number.

$-7+13=6\ \text{or}\ 19-13=6$

The midpoint value will be $6$ .

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Question

What is the midpoint of a real number line with points at and ?

Answer

Method A:

To find the midpoint, draw the number line that contains points and .

Then calculate the distance between the two points. In this case, the distance between and is . By dividing the distance between the two points by 2, you establish the distance from one point to the midpoint. Since the midpoint is 12 away from either end, the midpoint is 5.

Method B:

To find the midpoint, use the midpoint formula:

$M=\frac{x_{1}+x_{2}}{2}$

$M=\frac{-7+17}{2}$

$M=5$

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Question

A line segment has endpoints at and .

What is the midpoint of the line segment?

Answer

The midpoint can be found by using equations that calculate the values of its x- and y-coordinates. The x-coordinate can be found using the following equation:

$\frac{x_{1} + x_{2}}{2}$

Likewise, we will calculate the y-coordinate using another formula:

$\frac{y_{1} + y_{2}}{2}$

These formulas take the average of each coordinate separately in order to calculate the midpoint. In order to solve our question we will substitute in our given values and solve.

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

$\left (\frac{1+7}{2},\frac{-4+8}{2} \right )=\left (\frac{8}{2},\frac{4}{2} \right )$

$\left (\frac{8}{2},\frac{4}{2} \right )=(4,2)$

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Question

What is the midpoint of a line segment with endpoints and ?

Answer

The midpoint of a line can be found using the midpoint formula, given by:

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

Thus when we plug in our values we get the midpoint is

$\left(\frac{3+-1}{2},\frac{2+8}{2} \right )$

$=\left(\frac{2}{2},\frac{10}{2} \right )$

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Question

What is the midpoint of the line with endpoints:

$(10,-18)\textup{ and}$ ?

Answer

To find the midpoint given two points, use the formula:

$(\frac{x_1 +x_2}{2},\frac{y_1 +y_2}{2})$ . Thus for our points we see:

$(\frac{10+16}{2}, \frac{-18+8}{2})$ =

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Question

A school superintendent is observing bus routes of students using a town map overlaid with a coordinate plane. Two students, Jon and Steve, live on the same street. Their houses correspond to the coordinates of (-3, 4) and (5,8), respectively, on the map. If there is a bus stop exactly between their two houses, what are the coordinates of the bus stop on the map?

Answer

The midpoint formula must be used to find the midpoint of the line segment joining Jon's and Steve's houses. Use the following formula to find the midpoint:

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

In this formula, $(x_{1},y_{1})$ and $(x_{2}, y_{2})$ are the coordinates of the students' homes. Substitute in the coordinates of the houses and solve for the midpoint.

$\text{Midpoint}=\left(\frac{(-3)+5}{2},\frac{4+8}{2}\right)$

$\text{Midpoint}=\left(1,6\right)$

In this case, the bus stop is at the following coordinate on the town map:

$\left(1,6\right)$

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