Question

Find b such that the points A(2,b),B(5,5)and C(-6,0) are the vertices of right angled  with angle < BAC = 90 degree.​

Answer 1

• b = -3 or b = 8

Explanation:

We know that the perpendicular lines have negative-reciprocal slopes.

In other words, their product is - 1.

Find the slopes of AB and AC:

• m(AB) = (5 - b)/(5 - 2) = (5 - b) / 3
• m(AC) = (0 - b)/(- 6 - 2) = - b / - 8 = b/8

Find the the product of slopes and solve for b:

• (5 - b)/3 * b/8 = - 1
• b(b - 5)/24 = 1
• b(b - 5) = 24
• b² - 5b - 24 = 0
• b² - 8b + 3b - 24 = 0
• b(b - 8) + 3(b - 8) = 0
• (b + 3)(b - 8) = 0
• b = - 3 or b = 8