x,yWhat is the value of x and y if the point (x, y) is equidistant from (3, 6) and (4,8)?

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sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the of equations that give the coordinates x,y midpoint of segment, such that:

`{(x = (x_1 + x_2)/2),(y = (y_1 + y_2)/2):}`

Considering `(x_1,y_1) = (3,6)` and `(x_2,y_2) = (4,8)` yields:

`x = (3 + 4)/2 => x = 7/2 => x = 3.5`

`y = (6 + 8)/2 => y = 7`

Hence, evaluating the coordinates of the midpoint, under the given conditions, yields `x = 3.5, y = 7.`

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the point (x,y) is equidistant from the endpoints A(3, 6) and B(4,8), that means that the point is located on the midperpendicular of the segment AB.

The slope of the midperpendicular multiplied by the slope of the segment AB is -1.

mAB*m = -1

We'll write the equation of the segment line AB.

(yB - yA)/(y-yA) = (xB-xA)/(x-xA)

We'll substitute the coordinates of A and B:

(8-6)/(y-6) = (4-3)/(x-3)

2/(y-6) = 1/(x-3)

We'll cross multiply and we'll get:

y - 6 = 2(x-3)

The slope of the segment AB is mAB = 2.

mAB*m = -1

2*m = -1

m = -1/2

The equation of the midperpendicular is:

y - yM = (-1/2)(x-xM)

We'll calculate the coordinates of th midpoint of the segment AB:

xM = (xA+xB)/2

xM = (3+4)/2

xM = 7/2

yM = (yA+yB)/2

yM = (6+8)/2

yM = 7

y - 7 = (-1/2)(x - 7/2)

y - 7 = -x/2 + 7/4

4y - 28 = -2x + 7

2x - 4y - 35 = 0

The equation of the midperpendicular is: 2x - 4y - 35 = 0, so that the coordinates that verify the equation of the midperpendicular are equidistant from the endpoints A and B.

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