Mathematics — Analysis And Approaches

Mathematics — Analysis and Approaches

Problem

Let

a,b,c

be positive real numbers and let

P(a,0,0)

Q(0,b,0)

and

R(0,0,c)

be three points on the coordinate axes. a. (i) Express the side lengths

PQ,QR

and

RP

in terms of

a,b

and

c

. b. (ii) Prove that

\triangle PQR

is isosceles if and only if exactly two of the numbers

a,b,c

are equal. c. (iii) Show that

\triangle PQR

is equilateral if and only if

a=b=c

. d. (iv) Discuss under what circumstances (if any) the triangle becomes degenerate, giving reasons based on the side lengths obtained in part (i).