The points A (-5,4), B (-7,6) and C (5,2) are the co-ordinates of a right…

Answer: None of these — no angle of the triangle is a right angle.

Step 1: Compute vectors between the points.

• AB = B - A = (-7 - (-5), 6 - 4) = (-2, 2)

• AC = C - A = (5 - (-5), 2 - 4) = (10, -2)

• BC = C - B = (5 - (-7), 2 - 6) = (12, -4)

Step 2: Compute squared lengths of the sides.

• AB^2 = (-2)^2 + 2^2 = 4 + 4 = 8

• AC^2 = 10^2 + (-2)^2 = 100 + 4 = 104

• BC^2 = 12^2 + (-4)^2 = 144 + 16 = 160

Step 3: Check the Pythagorean relation for each possible right angle.

• Check if angle at A is right: AB^2 + AC^2 = 8 + 104 = 112, which is not equal to BC^2 = 160.

• Check if angle at B is right: BA^2 + BC^2 = 8 + 160 = 168, which is not equal to AC^2 = 104.

• Check if angle at C is right: CA^2 + CB^2 = 104 + 160 = 264, which is not equal to AB^2 = 8.

Optional check using dot products (adjacent side vectors):

• At A: AB · AC = (-2)*10 + 2*(-2) = -24 ≠ 0

• At B: BA · BC = 2*12 + (-2)*(-4) = 32 ≠ 0

• At C: CA · CB = (-10)*(-12) + 2*4 = 128 ≠ 0

Conclusion: None of the three angles satisfies the right-angle condition, so the correct answer is None of these.