Mathematics

Problem

Let

P(4,0)

and

Q(0,5)

be two points in the plane. A line

L_{1}

passes through

P

and is inclined at

45^{\circ}

to the positive

x

-axis. A second line

L_{2}

passes through

Q

and is perpendicular to

L_{1}

. a. (i) Find the equations of

L_{1}

and

L_{2}

. b. (ii) A line

L_{3}

is drawn through

Q

parallel to

L_{1}

and a line

L_{4}

is drawn through

P

parallel to

L_{2}

. Obtain the equations of

L_{3}

and

L_{4}

. c. (iii) The four lines

L_{1},L_{2},L_{3},L_{4}

enclose a quadrilateral. Prove that this quadrilateral is a rectangle and calculate its area.