Mathematics

Problem

Consider the right-angled triangle with vertices

A(0,0)

B(6,0)

and

C(0,8)

. A point

P(k,2)

, where

k

is a real parameter, is taken on the line

y=2

. a. (i) Obtain the exact numerical value of the area of

\triangle ABC

. b. (ii) Express, in terms of

k

, the areas of

\triangle PAB

\triangle PBC

and

\triangle PCA

. c. (iii) Using the results of part (ii), find the set of all real values of

k

for which the point

P

lies strictly inside

\triangle ABC

. d. (iv) Determine the value(s) of

k

, if any, for which

P

lies on one of the sides of the triangle.