Method 1 : distance formula. Use √[(x1 - x2)² + (y1 - y2)²]


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

2² + (y - 2)² = (-3)² + (y - 7)²

4 + y² - 4y + 4 = 9 + y² - 14y + 49

y² - 4y + 8 = y² - 14y + 58

-4y + 14y = 58 - 8

10y = 50
y = 5

Method 2 : Midpoint theorem

If (6,y) is equidistant from the two points, then we could draw a line from it to the midpoint of the line joining the two points.

The first line is the perpendicular bisector of the second. The product of their gradients must be -1.

Midpoint of (4,2) and (9,7)
Use formula ( (x1+x2)/2 , (y1+y2)/2 )

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

Gradient of line joining (4,2) and (9,7)
Use formula (y2-y1)/(x2-x1)

= (7 - 2)/(9 - 4)
= 5/5
= 1

Gradient of perpendicular bisector

= -1 ÷ 1
= -1


So -1 = (y-4.5)/(6-6.5)

-1 = (y-4.5)/-0.5

-1 × (-0.5) = y - 4.5
0.5 = y - 4.5

y = 0.5 + 4.5
y = 5

Method 3: just comparing gradients.

The gradient of the line joining (6,y) and (4,2) is the reciprocal of the gradient of the line joining (9,7) and (6,y)

Eg. Something like 4/5 and 5/4

Their product = 1

(We already know the lines are not horizontal or vertical so there won't be any 0 or undefined gradients)

So,

(7 - y)/(9 - 6) × (y - 2)/(6 - 4) = 1
(7 - y)/3 × (y - 2)/2 = 1
(7 - y)(y - 2)/ 6 = 1
(7 - y)(y - 2) = 6

7y - 14 - y² + 2y = 6
y² - 9y + 20 = 0
(y - 4)(y - 5) = 0

y = 4

(rejected as their rise and run would be 3/3 and 2/2. This means they are not equidistant from (6,y) as one line is longer than the other)

or y = 5