Question

M is the midpoint of AN, A has coordinates

Answer 1

Step-by-step explanation:

According to Question , M is the midpoint of AN, A has coordinates (-7,4), and M has coordinates (-4, -1).

Figure :-

$\setlength{\unitlength}{1 cm}\begin{picture}(20,12) \put(4,0.2){\line(0,-1){0.4}}\put(1,0){\line(1,0){6}} \put(3.8,-0.6){ \bf M(-4,-1) } \put(1,-0.6){ \bf A (-7,4) } \put(6.8,-0.6){ \bf N(x,y) } \end{picture}$

Let the coordinates of N be ( x , y )

Now , according to Midpoint Formula , the midpoint of points say A(x , y) and B (x' , y') is given by ,

$\boxed{\blue{ \bf Midpoint =\bigg( \dfrac{x+x'}{2} , \dfrac{y+y'}{2} \bigg) }}$

On using this formula ,

$=> Midpoint = \bigg( \dfrac{x+x'}{2} , \dfrac{y+y'}{2} \bigg) \\ \\ => (-4, 1) = \bigg( \dfrac{x -7 }{2} , \dfrac{y + 4 }{2}\bigg) \\ \\ => -4 = \dfrac{x-7}{2} \\\\ => x - 7 = -8 \\\\ => x = 7 -8 \\\\ \boxed{\red{\sf=> x = -1 }} \\\\ => \dfrac{y+4}{2}=-1 \\\\=> y +4 = -2 \\ \\ y = -4-2 \\\\\boxed{\red{\sf=> y = -6 }}$

Answer 2

Answer:   (-1, -6)

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Explanation:

Let's focus on the x coordinates of each point for now.

• Point A has x coordinate -7
• Point N has an unknown x coordinate. We leave it as x for now.
• Point M has x coordinate -4

The idea to find the midpoint's coordinates is to add up the endpoints coordinates and divide by 2.

So we'll add up -7 and x, then divide by 2 to get the midpoint x coordinate of -4

In short we have:  (-7+x)/2 = -4

Solving for x leads to...

(-7+x)/2 = -4

-7+x = 2*(-4)

-7+x = -8

x = -8+7

x = -1

This is the x coordinate of point N

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We'll repeat the same idea and steps for the y coordinates

• Point A has y coordinate 4
• Point N has y coordinate unknown. Leave it as y for now.
• Point M has y coordinate -1

So,

(4+y)/2 = -1

4+y = 2(-1)

4+y = -2

y = -2-4

y = -6

This is the y coordinate of point N.

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In summary, we found that point N is located at (-1, -6)

Point A is (-7,4) and point N is (-1,-6)

If you apply the midpoint formula to those two points, you should find the midpoint to be M(-4,-1) which helps confirm the answer.

Another way to confirm the answer is to compute the distance from A to M, which finds the length of segment AM. Do the same for segment MN. You should find that both segments are the same length.

The diagram below is a visual way to check. Starting at point A and moving down to point B, we moved 5 units. Then go from B to M and you'll move 3 units to the right. This pattern "down 5, right 3" is applied again when we go from M to C to N in that order. This effectively creates two congruent triangles (ABM and MCN) which can lead to proving AM = MN.

In short: the red piece of the graph is the same length as the blue piece; this confirms we have the correct answer.