Question

If A(4, 3), B(6, −2) and C(a, −3) are the vertices of a triangle right angled at A, find a.

Answer 1

ABC is right-angled at A, and the coordinates are:

A = (4, 3)

B = (6, −2)

C = (a, −3)

`vec(AB)` = B − A = (6 − 4, −2 − 3) = (2, −5)

`vec(AC)` = C − A = (a − 4, −3 − 3) = (a − 4, −6)

The dot product `vec(AB) · vec(AC)` = 0: (2)(a - 4) + (-5)(-6) = 0
2(a − 4) + 30 = 0
2a − 8 + 30 = 0
2a + 22 = 0
2a = −22
a = −11

Therefore, the value of a is −11.