Question

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?

Answer 1

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:

`{(x2 - x1)}^(2) + {(y2 - y1)}^(2) = {dist}^(2)`
So,

`{(3 - ( - 3))}^(2) + {( - 4 - 8)}^(2) =`

`{6}^(2) + {( - 12)}^(2) = 36 + 144 = 180`
and

`√(180) = √(36 * 5) = 6 √(5)`
so the distance is

`6 √(5)`
to find the missing y, we use the midpoint formula: (x,y) where

`x = x2 - (x2 - x1)/(2) \: \: y = y2 - (y2 - y1)/(2)`
we just need the y, so


`y = - 4 - ( - 4 - 8)/(2) = 2`
so y=2

now, the next question, we use the midpoint formula to find the midpoint between (7,8) and (-3,2)

`x = - 3 - ( - 3 - 7)/(2) = 2 \\ y = 2 - (2 - 8)/(2) = 5`
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.

`{(1 - 6)}^(2) + {9 - 8}^(2) = 25 - 1 = 24 \\ √(24) < 5`
so it is not A.

`{(1 - 2)}^(2) + {(9 - 4)}^(2) = 26 \\ √(26) > 5`
so it is not B

`{(1 - 3)}^(2) + {(9 - 3)}^(2) = 40 \\ √(40) > 5`
so it is not C

`{(1 - 5)}^(2) + {(9 - 6)}^(2) = 25 \\ √(25) = 5`
so the answer is D (5,6)

Finally, some algebra and the midpoint formula can give us y.

`y = y2 - (y2 - y1)/(2) \\ 11= 7 - (7 - y1)/(2)`
multiply everything by 2 to get rid of the denominator then combine like terms.

`11 * 2 = 7 * 2 - (7 - y1)/(2) * 2 \\ 22 = 14 - (7 - y1) \\ 22 = 14 - 7 + y1 \\ 22 = 7 + y1`

finally, the 7 is positive, so we need to subtract 7 from both sides.

`22 = 7 + y1 \\ 22 - 7 = 7 - 7 + y1 \\ 15 = y1`
so our missing y is 15.

hope that helps!