Chapter 9.5, Problem 9PSA

a.

To determine

Toverify: PQRS is an isosceles trapezoid.

Explanation of Solution

Given information:

A trapezoid has vertices at $P=\left(-a,0\right),Q=\left(-c,b\right),R=\left(c,b\right)$ and $S=\left(a,0\right)$ .

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoreantheorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Proof:

  

  $\begin{array}{l}PQ=\sqrt{{\left(-a+c\right)}^2+{\left(0-b\right)}^2}\\ =\sqrt{c^2-2ac+a^2+b^2}\\ SR=\sqrt{{\left(a-c\right)}^2+{\left(0-b\right)}^2}\\ =\sqrt{a^2-2ac+c^2+b^2}\end{array}$

  $PQ=SR$

PQRS is an isosceles trapezoid because two lengths are equal.

Therefore PQRS is an isosceles trapezoid.

b.

To determine

To prove:Diagonals $\overline{PR}$ and $\overline{QS}$ are congruent.

Explanation of Solution

Given information:

An trapezoid has vertices at $P=\left(-a,0\right),Q=\left(-c,b\right),R=\left(c,b\right)$ and $S=\left(a,0\right)$ .

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoreantheorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Proof:

  

  $\begin{array}{l}PR=\sqrt{{\left(c+a\right)}^2+{\left(b-0\right)}^2}\\ =\sqrt{c^2+2ac+a^2+b^2}\\ QS=\sqrt{{\left(a+c\right)}^2+{\left(0-b\right)}^2}\\ =\sqrt{c^2+2ac+a^2+b^2}\end{array}$

  $PR=QS$

  $\therefore PR\cong QS$

Diagonals $\overline{PR}$ and $\overline{QS}$ are congruent.