Giddy was correct.
It is easily understood if you consider two right-angled triangles. The first has a base of 5, height 12 and hypotenuse 13 (25 + 144 = 169, area 5 x 12 / 2 = 30) and the second a base of 9, height of 12 and hypotenuse of 15 (81 + 144 = 225 and area 9 x 12 / 2 = 54). Put them alongside each other, joined on the side of length 12, and you have a single triangle of base 5 + 9 = 14 and sides of 13 and 15 with a height of 12 and an area of 30 + 54 = 84.
It just goes to show how ignorant (or is that arrogant) some maths teachers are. BTW I have an Honours degree in Mathematics. "Those that can, do; those that can't, teach."