Chapter 1.8, Problem 116E

  $\left(a\right)$

To determine

To calculate: The new coordinates of the point $\left(5,3\right)$ when the coordinate plane is shifted.

Answer to Problem 116E

The new coordinates of the point $\left(5,3\right)$ are $\left(8,5\right)$ .

Explanation of Solution

Given information:

All points in the coordinate in plane are shifted 3 units to the right and 2 units upwards.

Formula used:

If $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

Calculation:

Consider the point $\left(5,3\right)$ . It is provided that all points in the coordinate in plane are shifted 3 units to the right and 2 units upwards that is shifted by the pair $\left(3,2\right)$ .

Recall that if $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

So, the point $\left(5,3\right)$ is shifted to,

  $\left(5,3\right)+\left(3,2\right)=\left(8,5\right)$

Thus, the new coordinates of the point $\left(5,3\right)$ are $\left(8,5\right)$ .

  $\left(b\right)$

To determine

To calculate: The new coordinates of the point $\left(a,b\right)$ when the coordinate plane is shifted.

Answer to Problem 116E

The new coordinates of the point $\left(a,b\right)$ are $\left(a+3,b+2\right)$ .

Explanation of Solution

Given information:

All points in the coordinate in plane are shifted 3 units to the right and 2 units upwards.

Formula used:

If $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

Calculation:

Consider the point $\left(a,b\right)$ . It is provided that all points in the coordinate in plane are shifted 3 units to the right and 2 units upwards that is shifted by the pair $\left(3,2\right)$ .

Recall that if $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

So, the point $\left(a,b\right)$ is shifted to,

  $\left(a,b\right)+\left(3,2\right)=\left(a+3,b+2\right)$

Thus, the new coordinates of the point $\left(a,b\right)$ are $\left(a+3,b+2\right)$ .

  $\left(c\right)$

To determine

To calculate: The old coordinates of the point $\left(3,4\right)$ when the coordinate plane is shifted.

Answer to Problem 116E

The old coordinates of the point $\left(3,4\right)$ are $\left(0,2\right)$ .

Explanation of Solution

Given information:

All points in the coordinate in plane are shifted 3 units to the right and 2 units upwards.

Formula used:

If $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

Calculation:

Consider the point $\left(3,4\right)$ . It is provided that all points in the coordinate in plane are shifted 3 units to the right and 2 units upwards that is shifted by the pair $\left(3,2\right)$ .

Recall that if $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

So, the point $\left(3,4\right)$ was originally at, let that point be $\left(x,y\right)$

  $\begin{array}{c}\left(x,y\right)+\left(3,2\right)=\left(3,4\right)\\ \left(x,y\right)=\left(3,4\right)-\left(3,2\right)\\ =\left(3-3,4-2\right)\\ =\left(0,2\right)\end{array}$

Thus, the old coordinates of the point $\left(3,4\right)$ are $\left(0,2\right)$ .

  $\left(d\right)$

To determine

To calculate: The new coordinates of the points $A,B\text{ and }C$ when the coordinate plane is shifted.

  

Answer to Problem 116E

The new coordinates of the points $A,B\text{ and }C$ are $A\text{'}=\left(-2,1\right),B\text{'}=\left(0,4\right)\text{ and }C\text{'}=\left(5,3\right)$ .

Explanation of Solution

Given information:

All points in the coordinate in plane are shifted 3 units to the right and 2 units upwards.

Formula used:

If $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

Calculation:

Consider the triangle $ABC$ shifted to triangle $A\text{'}B\text{'}C\text{'}$ .

  

It is provided that all points in the coordinate in plane are shifted 3 units to the right and 2 units upwards that is shifted by the pair $\left(3,2\right)$ .

Recall that if $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ are two pair of coordinates then $\left(x_1,y_1\right)+\left(x_2,y_2\right)=\left(x_1+x_2,y_1+y_2\right)$ .

So, the point $A\left(-5,-1\right)$ is shifted to,

  $\left(-5,-1\right)+\left(3,2\right)=\left(-2,1\right)$

So, the point $B\left(-3,2\right)$ is shifted to,

  $\left(-3,2\right)+\left(3,2\right)=\left(0,4\right)$

So, the point $C\left(2,1\right)$ is shifted to,

  $\left(2,1\right)+\left(3,2\right)=\left(5,3\right)$

Thus, the new coordinates of the points $A,B\text{ and }C$ are $A\text{'}=\left(-2,1\right),B\text{'}=\left(0,4\right)\text{ and }C\text{'}=\left(5,3\right)$ .