Chapter 13.9, Problem 6WE

a.

To determine

To show: that the given trapezoid is isosceles.

Explanation of Solution

Given information:

  $MNOP$ is a trapezoid with vertices $M(d,c),N(b,c),O(0,0),P(a,0)$ and $a=b+d$ .

  

Calculation:

We have to show $\overline{ON}=\overline{MP}$ . Now,

  $\begin{array}{l}\overline{ON}=\sqrt{{(b-0)}^2+{(c-0)}^2}=\sqrt{b^2+c^2}\\ \overline{MP}=\sqrt{{(a-d)}^2+c^2}=\sqrt{b^2+c^2}\text{=}\overline{ON}\text{ (Since }a-d=b)\end{array}$

Thus the trapezoid is isosceles.

b.

To determine

To show: that the diagonal of the trapezoid are congruent.

Explanation of Solution

Given information:

  $MNOP$ is a trapezoid with vertices $M(d,c),N(b,c),O(0,0),P(a,0)$ and $a=b+d$ .

  

Calculation:

Let us join the diagonal and we have to show $\overline{MO}=\overline{NP}$ . Now,

  $\begin{array}{l}\overline{NP}=\sqrt{{(a-b)}^2+{(c-0)}^2}=\sqrt{d^2+c^2}\text{ (Since }a-b=d)\\ \overline{MO}=\sqrt{{(d-0)}^2+{(c-0)}^2}=\sqrt{d^2+c^2}=\overline{NP}\end{array}$

Thus the diagonals are congruent to each other.