How To Find Corner Points Algebraically

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Point A is (five, seven). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

Possible Answers:

(29, –15)

(8.5, –ii)

(12, –11)

(22, –nine)

None of the other answers

Caption:

Bespeak A is (v, 7). Point B is (x, y). The midpoint of AB is (17, –4). What is the value of B?

We need to use our generalized midpoint formula:

MP = ( (5 + ten)/2, (7 + y)/two )

Solve each separately:

(5 + x)/2 = 17 → 5 + x = 34 → x = 29

(7 + y)/two = –4 → seven + y = –8 → y = –15

Therefore, B is (29, –15).

$\textup{Find the distance between the midpoint of the line segment}\\ \textup{ between points } (-1,-1)\textup{ and } (2,3) \textup{ and the midpoint of the line}\\\textup{ segment between points }(-1,-1)\textup{ and (2,-1).}$

Correct answer:

$\textup{None of the above.}$

Line segment Ac has ane endpoint at (-5,-2) . If this line's midpoint is at the origin, what are the coordinates of its other endpoint?

Correct answer:

(5,2)

Explanation:

A line'due south midpoint is the coordinate pair of that line which has the same number of points on either side of it. It bisects the line in ii equal parts.

Solution:

We are given that the line has an endpoint at (-5,-2) and its midpoint is on the origin. This known indicate would be in the Quadrant 3 and since on the reverse side of the midpoint at that place is exactly as much line we know that the other half of our line will lie in the Quadrant I. Add the absolute value of our known point to the coordinates of the origin to get (5,2) . This is the unknown endpoint. You should recognize that this end indicate is exactly the same altitude in the x and y direction (only opposite) as our given endpoint.

Line segment XY has a midpoint of (6,3) . If X is (2,8) what is Y?

Correct answer:

(10,-2)

Explanation:

For this kind of problem, it'southward important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, we'll have to solve this problem backwards. Since we've been given one of the endpoints (X) and nosotros just need to solve for the other end point (Y), we may arbitrarily assign (2,8) as (x_1,y_1) . If we substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

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

$(\frac{2+x_2}{2},\frac{8+y_2}{2}) = (6,3)$

Information technology may exist visually easier to break the arithmetic into separate operations.

$\frac{2+x_2}{2}=6$ and $\frac{8+y_2}{2}=3$

By separating the 10 and y components, we can hands solve for the missing endpoint now.

$2[\frac{2+x_2}{2}=6]$

2+x_2=12

$x_2={\color{Blue} 10}$

Doing similar arithmetics, y_2 will be solved to be .

Therefore, endpoint Y is (10,-2)

Line segment EF has a midpoint of (7,6) . If endpoint F is at (2,3) , what's the coordinate for endpoint Due east?

Correct answer:

(12,9)

Caption:

For this kind of trouble, it'south important to keep in mind how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Considering we've been given the midpoint already, we'll accept to solve this trouble backwards. Since nosotros've been given one of the endpoints (F) and we just need to solve for the other end betoken (East), we may arbitrarily assign (2,3) as (x_1,y_1) . If nosotros substitute in the two coordinates - an endpoint and the midpoint - we should be able to solve for the unknown endpoint.

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

$(\frac{2+x_2}{2},\frac{3+y_2}{2}) = (7,6)$

It may be visually easier to interruption the arithmetics into separate operations.

$\frac{2+x_2}{2}=7$ and $\frac{3+y_2}{2}=6$

By separating the x and y components, we tin easily solve for the missing endpoint at present.

$2[\frac{2+x_2}{2}=7]$

2+x_2=14

$x_2={\color{Blue} 12}$

Doing similar arithmetics, y_2 volition exist solved to be .

Therefore, endpoint Eastward is (12,9) .

Line segment DF has a midpoint of (4,8) . If endpoint D is at (22,7) , where is endpoint F?

Right answer:

(-14,9)

Explanation:

For this kind of problem, information technology's important to keep in listen how midpoint is solved for:

$midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) = (x_m,y_m)$ where is the midpoint coordinate.

Because we've been given the midpoint already, nosotros'll have to solve this problem backwards. Since nosotros've been given one of the endpoints (D) and we just need to solve for the other cease point (F), nosotros may arbitrarily assign (22,7) equally (x_1,y_1) . If we substitute in the two coordinates - an endpoint and the midpoint - nosotros should be able to solve for the unknown endpoint.

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

$(\frac{22+x_2}{2},\frac{7+y_2}{2}) = (4,8)$

Information technology may be visually easier to break the arithmetic into carve up operations.

$\frac{22+x_2}{2}=4$ and $\frac{7+y_2}{2}=8$

Past separating the x and y components, we can easily solve for the missing endpoint now.

$2[\frac{22+x_2}{2}=4]$

22+x_2=8

$x_2={\color{Blue} -14}$

Doing like arithmetics, y_2 will be solved to be .

Therefore, endpoint Y is (-14,9) .

The midpoint of a line segment is represented by the point $\left(\frac{15}{2},1\right)$ . If the coordinates for one of its endpoints are (2,10) and the y-coordinate of the other endpoint is 5, detect the value of the x-coordinate. To clarify, our endpoints are (2,10) and (?,5)

Right answer:

x=0

Caption:

We know that the midpoint of our line segment is $\left(\frac{15}{2},1\right)$ . To find the x-coordinate of this segment, nosotros work backwards, starting with our midpoint formula. In this case, we but need to use the midpoint formula to solve for the 10-coordinate, which looks like:

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

Next, multiply both sides of the equation by 2, which gives united states of america:

2=2-x , which means our missing x-coordinate is 0. So, the endpoints of our line segment are $(0,5)\&(2,10)$ .

If the midpoint of a line segment is (3, 4) and one endpoint is (-one, 2), find the other endpoint.

Possible Answers:

(4, 2)

(4, 6)

(7, six)

(2, six)

(3, 8)

Caption:

To solve, we volition using the midpoint formula and substitute what nosotros know. The midpoint formula is:

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

where (x_1, y_1) and (x_2, y_2) are the endpoints.

Now, here is what we know:

$\text{midpoint} = (3, 4)$

(x_1, y_1) = (-1, 2)

Here is what we are solving for

$(x_2, y_2) = \ ???$

So, we volition substitute. We get

$(3, 4) = (\frac{-1 + x_2}{2}, \frac{2 + y_2}{2})$

We tin divide this into parts. We know

$3 = \frac{-1 + x_2}{2}$

and

$4 = \frac{2 + y_2}{2}$

So, we tin solve for x_2 and y_2 to find the other endpoint.

$3 = \frac{-1 + x_2}{2}$

$3 \cdot 2 = \frac{-1 + x_2}{2} \cdot 2$

6 = -1 + x_2

6 + 1 = -1 + x_2 + 1

7 = x_2

x_2 = 7

$4 = \frac{2 + y_2}{2}$

$4 \cdot 2 = \frac{2 + y_2}{2} \cdot 2$

8 = 2 + y_2

8 - 2 = 2 + y_2 - 2

6 = y_2

y_2 = 6

This give us the point (7, 6) . Therefore, the other endpoint is (7, 6) .

A line segment has the midpoint (-2,3) . One end betoken of the line segment is located at (5,6) . What is the other cease signal?

Correct answer:

(-9,0)

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How To Find Corner Points Algebraically

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