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What is the distance between points (-3,8) and (3,-4) round to the nearest tenth.

The midpoint between points (-3,8) and (3,-4) is (0,y) what is y?

What is the midpoint between points (7,8) and (-3,2)?

Which point is a distance of 5 units from the points (1,9)?
A.(6,8)
B.(2,4)
C.(3,3)
D.(5,6)

The midpoint between points (x,y) and (3,7) is (4,11).What is y?

Respuesta :

barryswindler
The distance between two points can be found by looking at the distance as the hypotenuse of a right triangle, where the bases are the distance between the x values (movement from left to right) and the y values (movement up or down). The formula looks like this:
[tex] {(x2 - x1)}^{2} + {(y2 - y1)}^{2} = {dist}^{2} [/tex]
So,
[tex] {(3 - ( - 3))}^{2} + {( - 4 - 8)}^{2} = [/tex]
[tex] {6}^{2} + {( - 12)}^{2} = 36 + 144 = 180[/tex]
and
[tex] \sqrt{180} = \sqrt{36 \times 5} = 6 \sqrt{5} [/tex]
so the distance is
[tex]6 \sqrt{5} [/tex]
to find the missing y, we use the midpoint formula: (x,y) where
[tex]x = x2 - \frac{x2 - x1}{2} \: \: y = y2 - \frac{y2 - y1}{2} [/tex]
we just need the y, so

[tex]y = - 4 - \frac{ - 4 - 8}{2} = 2[/tex]
so y=2

now, the next question, we use the midpoint formula to find the midpoint between (7,8) and (-3,2)
[tex]x = - 3 - \frac{ - 3 - 7}{2} = 2 \\ y = 2 - \frac{2 - 8}{2} = 5[/tex]
so our midpoint is at (2,5)

The next question is a bit more complicated. We use the distance formula and substitute in every point until we get an answer of 5.
[tex] {(1 - 6)}^{2} + {9 - 8}^{2} = 25 - 1 = 24 \\ \sqrt{24} < 5[/tex]
so it is not A.
[tex] {(1 - 2)}^{2} + {(9 - 4)}^{2} = 26 \\ \sqrt{26} > 5[/tex]
so it is not B
[tex] {(1 - 3)}^{2} + {(9 - 3)}^{2} = 40 \\ \sqrt{40} > 5[/tex]
so it is not C
[tex] {(1 - 5)}^{2} + {(9 - 6)}^{2} = 25 \\ \sqrt{25} = 5[/tex]
so the answer is D (5,6)

Finally, some algebra and the midpoint formula can give us y.
[tex]y = y2 - \frac{y2 - y1}{2} \\ 11= 7 - \frac{7 - y1}{2} [/tex]
multiply everything by 2 to get rid of the denominator then combine like terms.
[tex]11 \times 2 = 7 \times 2 - \frac{7 - y1}{2} \times 2 \\ 22 = 14 - (7 - y1) \\ 22 = 14 - 7 + y1 \\ 22 = 7 + y1[/tex]

finally, the 7 is positive, so we need to subtract 7 from both sides.
[tex]22 = 7 + y1 \\ 22 - 7 = 7 - 7 + y1 \\ 15 = y1[/tex]
so our missing y is 15.

hope that helps!


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