SOLUTION: How would you calculate the following problem???? Find the length of AB there is aline above it and the midpoint of ab 1) A (-10,6) 2) A (8,3) B (4,2)

Algebra ->  Algebra  -> Points-lines-and-rays -> SOLUTION: How would you calculate the following problem???? Find the length of AB there is aline above it and the midpoint of ab 1) A (-10,6) 2) A (8,3) B (4,2)       Log On

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Question 98441: How would you calculate the following problem????
Find the length of AB there is aline above it and the midpoint of ab
1) A (-10,6) 2) A (8,3)
B (4,2) B (7,-2)
Thanks,
Amber
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
1)

"Find the length of AB"
Start with the given distance formula
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 where is the first point and is the second point

d=sqrt%28%28-10-4%29%5E2%2B%286-2%29%5E2%29 Plug in x%5B1%5D=-10, x%5B2%5D=4, y%5B1%5D=6, y%5B2%5D=2

d=sqrt%28%28-14%29%5E2%2B%284%29%5E2%29 Evaluate -10-4 to get -14. Evaluate 6-2 to get 4.

d=sqrt%28196%2B16%29 Square each value

d=sqrt%28212%29 Add

d=2%2Asqrt%2853%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)


So the distance approximates to

d=14.560219778561

which rounds to
14.56

So the distance between (-10,6) and (4,2) is approximately 14.56 units


-----------------------------------------------------------------


"Find the midpoint of ab"


In order to find the midpoint, we need to average each corresponding coordinate. In other words, we need to add up the corresponding coordinates and divide the sum by 2.


So lets find the averages between the two points



To find , average the x-coordinates between the two points
x%5Bmid%5D=%28-10%2B4%29%2F2=%28-6%29%2F2=-3


So the x-coordinate of the midpoint is -3 (i.e. x=-3)
-----------------------------------------------------------------------------------------------------------------


To find , average the y-coordinates between the two points
y%5Bmid%5D=%286%2B2%29%2F2=%288%29%2F2=4


So the y-coordinate of the midpoint is 4 (i.e. y=4)
-----------------------------------------------------------------------------------------------------------------
Answer:
Since the coordinates of the midpoint are x=-3, y=4, this means the midpoint is (-3,4)

Check:
Here is a graph to visually see the answer
drawing%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C%0D%0Agraph%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C0%29%2C%0D%0Ablue%28line%28-10%2C6%2C4%2C2%29%29%2C%0D%0Acircle%28-10%2C6%2C0.05%29%2C%0D%0Acircle%28-10%2C6%2C0.08%29%2C%0D%0Acircle%284%2C2%2C0.05%29%2C%0D%0Acircle%284%2C2%2C0.08%29%2C%0D%0Acircle%28-3%2C4%2C0.05%29%2C%0D%0Acircle%28-3%2C4%2C0.08%29%0D%0A%0D%0A%29 Graph of the line segment with the endpoints (-10,6) and (4,2) with the midpoint (-3,4)



We could visually verify our answer if we simply draw right triangles from each point like this:
drawing%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C%0D%0Agraph%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C0%29%2C%0D%0Ablue%28line%28-10%2C6%2C4%2C2%29%29%2C%0D%0A%0D%0Aline%28-10%2C6%2C-10%2C%286%2B2%29%2F2%29%2C%0D%0Aline%28-10%2C%286%2B2%29%2F2%2C-3%2C4%29%2C%0D%0Aline%28-3%2C4%2C%28-10%2B4%29%2F2%2C2%29%2C%0D%0Aline%28%28-10%2B4%29%2F2%2C2%2C4%2C2%29%2C%0D%0A%0D%0Acircle%28-10%2C6%2C0.05%29%2C%0D%0Acircle%28-10%2C6%2C0.08%29%2C%0D%0Acircle%284%2C2%2C0.05%29%2C%0D%0Acircle%284%2C2%2C0.08%29%2C%0D%0Acircle%28-3%2C4%2C0.05%29%2C%0D%0Acircle%28-3%2C4%2C0.08%29%0D%0A%0D%0A%29

Here we can see that the two triangles are congruent, so our answer is verified.






2)

"Find the length of AB"


Start with the given distance formula
d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 where is the first point and is the second point

d=sqrt%28%288-7%29%5E2%2B%283--2%29%5E2%29 Plug in x%5B1%5D=8, x%5B2%5D=7, y%5B1%5D=3, y%5B2%5D=-2

d=sqrt%28%281%29%5E2%2B%285%29%5E2%29 Evaluate 8-7 to get 1. Evaluate 3--2 to get 5.

d=sqrt%281%2B25%29 Square each value

d=sqrt%2826%29 Add

So the distance approximates to

d=5.09901951359278

which rounds to
5.1

So the distance between (8,3) and (7,-2) is approximately 5.1 units


------------------------------------------------------------------------------


"Find the midpoint of ab"


In order to find the midpoint, we need to average each corresponding coordinate. In other words, we need to add up the corresponding coordinates and divide the sum by 2.


So lets find the averages between the two points



To find , average the x-coordinates between the two points
x%5Bmid%5D=%288%2B7%29%2F2=%2815%29%2F2=7.5


So the x-coordinate of the midpoint is 7.5 (i.e. x=7.5)
-----------------------------------------------------------------------------------------------------------------


To find , average the y-coordinates between the two points
y%5Bmid%5D=%283%2B-2%29%2F2=%281%29%2F2=0.5


So the y-coordinate of the midpoint is 0.5 (i.e. y=0.5)
-----------------------------------------------------------------------------------------------------------------
Answer:
Since the coordinates of the midpoint are x=7.5, y=0.5, this means the midpoint is (7.5,0.5)

Check:
Here is a graph to visually see the answer
drawing%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C%0D%0Agraph%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C0%29%2C%0D%0Ablue%28line%288%2C3%2C7%2C-2%29%29%2C%0D%0Acircle%288%2C3%2C0.05%29%2C%0D%0Acircle%288%2C3%2C0.08%29%2C%0D%0Acircle%287%2C-2%2C0.05%29%2C%0D%0Acircle%287%2C-2%2C0.08%29%2C%0D%0Acircle%287.5%2C0.5%2C0.05%29%2C%0D%0Acircle%287.5%2C0.5%2C0.08%29%0D%0A%0D%0A%29 Graph of the line segment with the endpoints (8,3) and (7,-2) with the midpoint (7.5,0.5)



We could visually verify our answer if we simply draw right triangles from each point like this:
drawing%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C%0D%0Agraph%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C0%29%2C%0D%0Ablue%28line%288%2C3%2C7%2C-2%29%29%2C%0D%0A%0D%0Aline%288%2C3%2C8%2C%283%2B-2%29%2F2%29%2C%0D%0Aline%288%2C%283%2B-2%29%2F2%2C7.5%2C0.5%29%2C%0D%0Aline%287.5%2C0.5%2C%288%2B7%29%2F2%2C-2%29%2C%0D%0Aline%28%288%2B7%29%2F2%2C-2%2C7%2C-2%29%2C%0D%0A%0D%0Acircle%288%2C3%2C0.05%29%2C%0D%0Acircle%288%2C3%2C0.08%29%2C%0D%0Acircle%287%2C-2%2C0.05%29%2C%0D%0Acircle%287%2C-2%2C0.08%29%2C%0D%0Acircle%287.5%2C0.5%2C0.05%29%2C%0D%0Acircle%287.5%2C0.5%2C0.08%29%0D%0A%0D%0A%29

Here we can see that the two triangles are congruent, so our answer is verified.