Method 1 :

If (6,y) is equidistant from (4,2) and (9,7),
Then the length of (6,y) to (4,2) = length of (6,y) to (9,7)

Length of straight line between 2 points
= √[(x1-x2)² + (y1-y2)²]

So,

√[(6-4)² + (y-2)²] = √[(9-6)² + (7-y)²]

Square both sides,

2² + (y-2)² = 3² + (7-y)²

4 + y² - 4y + 4 = 9 + 49 - 14y + y²

8 - 4y = 58 - 14y

-4y + 14y = 58 - 8

10y = 50

y = 5

Method 2 :


If (6,y) is equidistant from (4,2) and (9,7), then the midpoint of (4,2) and (9,7) is the foot of the perpendicular from (6,y) to the line joining (4,2) and (9,7)

Midpoint = ( (4+9)/2 , (2+7)/2 )
= (13/2, 9/2)
= (6.5, 4.5)

Next, since they are perpendicular,
the product of the gradients of the line joining (4,2) and (9,7) and the line joining (6,y) and (6.5,4.5) is -1


(4.5 - y)/(6.5 - 6) x (7 - 2)/(9 - 4) = -1

(4.5 - y)/0.5 x 5/5 = -1

(4.5 - y)/0.5 = -1

4.5 - y = -0.5

y = 4.5 + 0.5

y = 5