Chapter 13.1, Problem 7CE

a.

To determine

To find the coordinates of T

Answer to Problem 7CE

The value of T is $\left(2,-1\right)$

Explanation of Solution

Looking at the graph given in the question

The given point is

  $\left(-4,-1\right)\text{ and }\left(2,-3\right)$

Calculation:

The coordinates of T can be obtained as

The point T lies on the horizontal line of $\left(-4,-1\right)$ and vertical line of $\left(2,-3\right)$

Let, $\left(x_1,y_1\right)\text{ be }\left(-4,-1\right)\text{ and }\left(x_2,y_2\right)\text{ be }\left(2,-3\right)$

Then the point T is $\left(x_2,y_1\right)=\left(2,-1\right)$

Hence,

The value of T is $\left(2,-1\right)$

b.

To determine

To find the length of the legs of the right triangle

Answer to Problem 7CE

The value of legs are $\left(6,2\right)$

Explanation of Solution

Looking at the graph given in the question

The given point is

  $\left(-4,-1\right)\text{ and }\left(2,-3\right)$

Calculation:

The length of the legs of the right triangle can be obtained as

The point T lies on the horizontal line of $\left(-4,-1\right)$ and vertical line of $\left(2,-3\right)$

Let, $\left(x_1,y_1\right)\text{ be }\left(-4,-1\right)\text{ and }\left(x_2,y_2\right)\text{ be }\left(2,-3\right)$

Then,

  $\begin{array}{l}\left(x_2-x_1\right)=\left(2-\left(-4\right)\right)\\ \left(x_2-x_1\right)=\left(2+4\right)\\ \left(x_2-x_1\right)=6\end{array}$

And

  $\begin{array}{l}\left(y_2-y_1\right)=\left(-3-\left(-1\right)\right)\\ \left(y_2-y_1\right)=\left(-3+1\right)\\ \left(y_2-y_1\right)=-2\end{array}$

The length of the legs cannot be negative

So, $\left(y_2-y_1\right)=2$

Hence,

The value of legs are $\left(6,2\right)$

c.

To determine

To find the length of the segment

Answer to Problem 7CE

The value of the length of the segment are $\sqrt{40}$

Explanation of Solution

Looking at the graph given in the question

The given point is

  $\left(-4,-1\right)\text{ and }\left(2,-3\right)$

Calculation:

The length of the segment can be obtained as

The point T lies on the horizontal line of $\left(-4,-1\right)$ and vertical line of $\left(2,-3\right)$

Let, $\left(x_1,y_1\right)\text{ be }\left(-4,-1\right)\text{ and }\left(x_2,y_2\right)\text{ be }\left(2,-3\right)$

Use the distance formula between the two points to find the segment

  $d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\left[d=\text{distance}\right]$

Put the value of $x_1=-4,x_2=2,y_1=-1\text{ and }{y}_2=-3$ on the above equation

  $\begin{array}{l}d=\sqrt{{\left(2-\left(-4\right)\right)}^2+{\left(-3-\left(-1\right)\right)}^2}\\ d=\sqrt{{\left(2+4\right)}^2+{\left(-3+1\right)}^2}\\ d=\sqrt{{\left(6\right)}^2+{\left(-2\right)}^2}\\ d=\sqrt{36+4}\\ d=\sqrt{40}\end{array}$

Hence,

The value of the length of the segment are $\sqrt{40}$