Mathematics

Problem

Let

\triangle ABC

have vertices

A(a,2)

B(3,b)

and

C(5,7)

. It is known that the area of the triangle is

6

square units. a. (i) Form an equation in

a

and

b

using the given information. b. (ii) If

a=1

, determine all possible values of

b

. c. (iii) For each value of

b

obtained in part (ii), decide whether

\triangle ABC

is acute–angled, right–angled or obtuse–angled. Justify your answer by calculating the squares of its side lengths.