Chapter 13.5, Problem 17WE

a.

To determine

To find: Verify that $\overline{OQ}$ and $\overline{PR}$ have the same midpoint.

Explanation of Solution

Given information:In figure, a quadrilateral $OPQR$ in which coordinates of points are $O(0,0),P(2,6),Q(9,9)\text{ and }R(7,3)$

Formula Used:If $A(x_1,y_1)\text{ and }B(x_2,y_2)$ then coordinates of midpoint of $\overline{AB}$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

Calculation:

Let midpoints of $\overline{OQ}$ and $\overline{PR}$ are S and T.

Coordinates of S is $\left(\frac{2+7}{2},\frac{6+3}{2}\right)\equiv \left(\frac{9}{2},\frac{9}{2}\right)$

Coordinates of T is $\left(\frac{0+9}{2},\frac{0+9}{2}\right)\equiv \left(\frac{9}{2},\frac{9}{2}\right)$

Since both of the midpoints is $\left(\frac{9}{2},\frac{9}{2}\right)$

So, $\overline{OQ}$ and $\overline{PR}$ have the same midpoint.

Hence verified.

b.

To determine

To find: $OPQR$ is $\underset{\_}{\text{ ? }}$ .

Answer to Problem 17WE

  $OPQR$ is a parallelogram.

Explanation of Solution

Given information:Part (a) shows that diagonals of $OPQR$ bisects each other.

Calculation:

From part (a):

Here, $\overline{OQ}$ and $\overline{PR}$ are diagonals $OPQR$ .

Since, both diagonals has same midpoint.

It means that both diagonals intersect each other at $\left(\frac{9}{2},\frac{9}{2}\right)$ which is midpoint.

So, diagonals of $OPQR$ bisects each other.

If diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Therefore, $OPQR$ is a parallelogram.

c.

To determine

To find: Use slopes to verify that the opposite sides of $OPQR$ are parallel.

Explanation of Solution

Given information:In figure, a quadrilateral $OPQR$ in which coordinates of points are $O(0,0),P(2,6),Q(9,9)\text{ and }R(7,3)$

Formula used: Theslope of a line passing through point $A(x_1,y_1)$ and $B(x_2,y_2)$ is $\frac{y_2-y_1}{x_2-x_1}$

Concept used: If slopes of two lines are equal then lines are parallel.

Calculation:

Here, sides OP and RQ, and PQ and OR are opposite sides.

  $\begin{array}{l}\text{Slope of }\overline{OP}=\frac{6-0}{2-0}\\ \Rightarrow \text{Slope of }\overline{OP}=3\\ \text{Slope of }\overline{RQ}=\frac{9-3}{9-7}\\ \Rightarrow \text{Slope of }\overline{RQ}=\frac{6}{2}\\ \Rightarrow \text{Slope of }\overline{RQ}=3\end{array}$

Since, slopes of $\overline{OP}$ and slope of $\overline{RQ}$ are equal.

So, sides OP and RQ are parallel.

  $\begin{array}{l}\text{Slope of }\overline{PQ}=\frac{9-6}{9-2}\\ \Rightarrow \text{Slope of }\overline{PQ}=\frac{3}{7}\\ \text{Slope of }\overline{OR}=\frac{3-0}{7-0}\\ \Rightarrow \text{Slope of }\overline{OR}=\frac{3}{7}\end{array}$

Since, slopes of $\overline{PQ}$ and slope of $\overline{OR}$ are equal.

So, sides PQ and OR are parallel.

Hence, opposite sides of $OPQR$ are parallel.

d.

To determine

To find: Use the distance formula to verify that the opposite sides are congruent.

Explanation of Solution

Given information:In figure, a quadrilateral $OPQR$ in which coordinates of points are $O(0,0),P(2,6),Q(9,9)\text{ and }R(7,3)$

Formula used: The distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by:

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

Concept used: If two sides are equal in length then they are congruent.

Calculation:

Here, sides OP and RQ, and PQ and OR are opposite sides.

  $\begin{array}{l}{d}_{OP}=\sqrt{{\left(2-0\right)}^2+{\left(6-0\right)}^2}\\ \Rightarrow d_{OP}=\sqrt{{\left(2\right)}^2+{\left(6\right)}^2}\\ \Rightarrow d_{OP}=\sqrt{4+36}\\ \Rightarrow d_{OP}=\sqrt{40}\\ \Rightarrow d_{OP}=2\sqrt{10}\text{ units}\\ d_{RQ}=\sqrt{{\left(9-7\right)}^2+{\left(9-3\right)}^2}\\ \Rightarrow d_{RQ}=\sqrt{{\left(2\right)}^2+{\left(6\right)}^2}\\ \Rightarrow d_{RQ}=\sqrt{4+36}\\ \Rightarrow d_{RQ}=\sqrt{40}\\ \Rightarrow d_{RQ}=2\sqrt{10}\text{ units}\end{array}$

Since, length of OP and RQ are equal.

So, sides OP and RQ are congruent.

  $OPQR$

Since, length of PQ and OR are equal.

So, sides PQ and ORare congruent.

Hence, opposite sides of $OPQR$ are congruent.