Chapter 9.5, Problem 8PSA

a.

To determine

Tofind: The coordinates of A , B , C and O .

Answer to Problem 8PSA

The coordinates are $A\left(0,2b\right),B\left(2a,2b\right),C\left(2a,0\right)$ and $O(0,0)$ .

Explanation of Solution

Given information:

In Rectangle ABCO , M is the midpoint of OA , N is midpoint of AB , P is the midpoint of BC and Q is midpoint of OC .

Calculation:

  

By inspection, we get$A\left(0,2b\right),B\left(2a,2b\right),C\left(2a,0\right)$ and $O(0,0)$ .

b.

To determine

To calculate: The coordinates of M , N , P and Q .

Answer to Problem 8PSA

The coordinates are $M\left(0,b\right),N\left(a,2b\right),P\left(2a,b\right)$ and $Q(a,0)$ .

Explanation of Solution

Given information:

In Rectangle ABCO , M is the midpoint of OA , N is midpoint of AB , P is the midpoint of BC and Q is midpoint of OC .

Calculation:

  

By inspection, we get

  $M=(\frac{0+0}{2},\frac{2b+0}{2})=\left(0,b\right)$

  $\begin{array}{l}P=(\frac{2a+2a}{2},\frac{2b+0}{2})=\left(2a,b\right)\\ N=(\frac{2a+0}{2},\frac{2b+2b}{2})=\left(a,2b\right)\\ Q=(\frac{2a+0}{2},\frac{0+0}{2})=\left(a,0\right)\end{array}$

c.

To determine

To find: The slopes of $\overline{MN},\overline{QP},\overline{MQ}$ and $\overline{NP}$ , and draw a conclusion about MNPQ.

Answer to Problem 8PSA

MNPQ is a parallelogram.

Explanation of Solution

Given information:

In Rectangle ABCO , M is the midpoint of OA , N is midpoint of AB , P is the midpoint of BC and Q is midpoint of OC .

Formula used:

Slope of a line:

  $m=\frac{y_2-y_1}{x_2-x_1}$

  $m=$ slope.

  $(x_1,y_1)=$ coordinates of first point in the line.

  $(x_2,y_2)=$ coordinates of second point in the line.

Calculation:

  

Slope of $\overline{MN}=\frac{2b-b}{a-0}=\frac{b}{a}$

Slope of $\overline{MQ}=\frac{0-b}{a-0}=-\frac{b}{a}$

Slope of $\overline{QP}=\frac{b-0}{2a-a}=\frac{b}{a}$

Slope of $\overline{NP}=\frac{2b-b}{a-2a}=-\frac{b}{a}$

Since MN is parallel to QP and MQ is parallel to NP , MNPQ is a parallelogram.

d.

To determine

To find: The lengths of $\overline{MN},\overline{QP},\overline{MQ}$ and $\overline{NP}$ , and draw a conclusion about MNPQ.

Answer to Problem 8PSA

MNPQ is a rhombus.

Explanation of Solution

Given information:

In Rectangle ABCO , M is the midpoint of OA , N is midpoint of AB , P is the midpoint of BC and Q is midpoint of OC .

Formula used:

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

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

Calculation:

  

  $\begin{array}{l}MN=\sqrt{{\left(a-0\right)}^2+{\left(2b-b\right)}^2}=\sqrt{a^2+b^2}\\ QP=\sqrt{{\left(2a-a\right)}^2+{\left(0-b\right)}^2}=\sqrt{a^2+b^2}\\ MQ=\sqrt{{\left(a-0\right)}^2+{\left(0-b\right)}^2}=\sqrt{a^2+b^2}\\ NP=\sqrt{{\left(2a-a\right)}^2+{\left(b-2b\right)}^2}=\sqrt{a^2+b^2}\end{array}$

  $MN=QP=MQ=NP$

→ MNPQ is a rhombus.