Given:
Vertices of the triangle: A(5,3), B(11,-5), P(12,y)

To determine whether the triangle is right-angled, we can use the Pythagorean theorem. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Step 1: Calculate the Length of AB:
Using the distance formula, we can find the length of AB:
AB = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(11 - 5)^2 + (-5 - 3)^2]
= √[6^2 + (-8)^2]
= √[36 + 64]
= √100
= 10

Step 2: Calculate the Length of AP:
Using the distance formula, we can find the length of AP:
AP = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(12 - 5)^2 + (y - 3)^2]
= √[7^2 + (y - 3)^2]

Step 3: Calculate the Length of BP:
Using the distance formula, we can find the length of BP:
BP = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(12 - 11)^2 + (y - (-5))^2]
= √[1^2 + (y + 5)^2]

Step 4: Apply the Pythagorean Theorem:
In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides:
AB^2 = AP^2 + BP^2

Substituting the values we calculated:
10^2 = [7^2 + (y - 3)^2] + [1^2 + (y + 5)^2]
100 = 49 + (y - 3)^2 + 1 + (y + 5)^2
100 = 50 + (y - 3)^2 + (y + 5)^2
50 = (y - 3)^2 + (y + 5)^2

Step 5: Solve the Equation:
Expanding the equation:
50 = y^2 - 6y + 9 + y^2 + 10y + 25
0 = 2y^2 + 4y + 16

Simplifying the equation:
y^2 + 2y + 8 = 0

Step 6: Solve for y:
Using the quadratic formula, we have:
y = (-b ± √(b^2 - 4ac)) / (2a)
y = (-2 ± √(2^2 - 4(1)(8))) / (2(1))
y = (-2 ± √(4 - 32)) / 2
y = (-2 ± √(-28)) / 2
y = (-2 ± 2