# A line segment with length 65 has one endpoint at (3, 9) . The other endpoint is at (-4, y) . Determine a value of y.

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We have a segment with end points (3,9) and (-4,y) . The segment is 65 unit long.

We know that the formula for distance between two points is:

d= sqrt[(x2-x1)^2 + (y2-y1)^2]

65= sqrt[(-4-3)^2 + (y-9)^2]

Square both sides:

4225= 49+ (y-9)^2

4176= (y-9)^2

64.6= y-9

==> y= 73.6 (approx.)

OR:

64.6= -(y-9)

==> y= -64.6+9= -55.6

To solve this we use distance formula, where the distance is known , and the y coordinate is determined by solving the equation.

The distance d between the two points (x1,y1) and (x2,y2) is d = sqrt{(x2-x1)^2+(y2-y1)^2}. Here given are d =65, (x1,y1) = (3,9) and (x2,y2) = (-4,y). Substitutig these in the formula,

65 = (sqrt{(-4-3)^2+(y-9)^2}. Squaring,

65^2 = (-7)^2+(y-9)^2.

4225-49 = (y-9)^2.

4176 = (y-9)^2. Taking square root,

+or- sqrt4176 = y-9. Or

y1 = 9+srt4176 = 73.622 nearly Or y2 = 9-sqrt4176 = -55.622.

So the values of y1 and y2 as above are the possible coordinates.

Length of a line connecting two points (x1, y1) and (x2, y2) is given by

Length = [x2 - x2)^2 + (y2 - y1)^2]^1/2

Substituting the coordinates of the two given points in the above formula, we get the length of given line as:

Length = [(-4 - 3)^2 + (y -9)^2]^1/2

= [(-7)^2 + (y - 9)^2]^1/2

= [49 + (y - 9)^2]^1/2

As length of the line is given to be equal to 65:

[49 + (y - 9)^2]^1/2 = 65

squaring both sides of the equation we get

49 + (y - 9)^2 = 4225

(y - 9)^2 = 4225 - 49 = 4176

y - 9 = 4176^(1/2)

y - 9 = 64.622 or (-64.622)

Taking positive value of 4176^(1/2)

y - 9 = 64.622

y = 64.622 + 9 = 73.622

Taking negative value of 4176^(1/2)

y - 9 = - 64.622

y = -64.622 + 9 = - 55.622

We need to calculate the length of a segment using the formula:

65 = sqrt[(3-(-4))^2 + (9-y)^2]

65 = sqrt(49 + 81 - 18y + y^2)

65 = sqrt (130 - 18y + y^2)

Now, we'll square raise both side:

65^2 = 130 - 18y + y^2

We'll subtract 4225 both sides:

y^2 - 18y + 130 - 4225 = 0

y^2 - 18y - 4095 = 0

We'll apply quadratic formula:

y1 = [18+sqrt(324+16380)]/2

y1 = (18+129.24)/2

y1 = 73.62

y2 = (18-129.24)/2

y2 = -55.62