The points and are two vertices of a

MATHEMATICS · Multi-part · AoPS · Mathematics · One-Variable Linear Equations · One-Variable Linear Equations

Problem

The points

P(1,2)

and

Q(4,6)

are two vertices of a

\triangle PQR

. The third vertex

R

lies on the line

y = -1

and has coordinates

R(k,-1)

. (i) Express the perimeter of

\triangle PQR

in terms of

k

. (ii) Given that the perimeter of

\triangle PQR

15\,\text{units}

, form an equation in

k

and solve it. (iii) For each valid value of

k

obtained in part (ii), write the corresponding coordinates of

R

and state the perimeter of

\triangle PQR

to two decimal places (verify your result).