Chapter 12.7, Problem 15E

a.

To determine

To draw a triangle ABC with the given vertices and find the coordinates after dilation and draw the final image.

Answer to Problem 15E

The final image after dilation is:

  

Explanation of Solution

Given information:

The vertices of the triangle ABC given are:

  $\text{A}\left(-2,4\right),\text{B}\left(4,0\right)\text{ and C}\left(2,-4\right)$

Dilate the triangle using a scale factor of $0.25$ and then dilate the image using a scale factor of $2$ .

Calculation :

The triangle ABC can be drawn as:

  

The coordinates after dilation using a scale factor of $0.25$ are:

  $\begin{array}{l}\text{A}\left(-2,4\right)\to \text{A'}\left(-0.5,1\right)\\ \text{B}\left(4,0\right)\to \text{B'}\left(1,0\right)\\ \text{C}\left(2,-4\right)\to \text{C'}\left(0.5,-1\right)\end{array}$

Hence, the coordinates using scale factor $0.25$ are $\text{A'}\left(-0.5,1\right),\text{B'}\left(1,0\right)\text{ and C'}\left(0.5,-1\right)$ .

The coordinates after dilating the image using a scale factor of $2$ :

  $\begin{array}{l}\text{A'}\left(-0.5,1\right)\to \text{A"}\left(-1,2\right)\\ \text{B'}\left(1,0\right)\to \text{B"}\left(2,0\right)\\ \text{C'}\left(0.5,-1\right)\to \text{C"}\left(1,-2\right)\end{array}$

Hence, the coordinates using scale factor $2$ are $\text{A''}\left(-1,2\right),\text{B''}\left(2,0\right)\text{ and C''}\left(1,-2\right)$ .

The final image, triangle $\text{A"B"C"}$ can be drawn by:

  

b.

To determine

To find the scale factor that could be used to dilate the triangle ABC into the final image triangle $\text{A"B"C"}$

Answer to Problem 15E

The scale factor to dilate the triangle ABC to the final image in one step is $0.5$

Explanation of Solution

Given information:

The vertices of the triangle ABC given are:

  $\text{A}\left(-2,4\right),\text{B}\left(4,0\right)\text{ and C}\left(2,-4\right)$

On dilating the triangle using a scale factor of $0.25$ and then dilating the image using a scale factor of $2$ , the coordinates are $\text{A''}\left(-1,2\right),\text{B''}\left(2,0\right)\text{ and C''}\left(1,-2\right)$ .

Calculation :

The above scale factors can be multiplied to obtain a single dilating factor for the final image.

  $2\times 0.25=0.5$

Hence, the scale factor to dilate the triangle ABC to the final image in one step is $0.5$

c.

To determine

To find the changes in final image if any, if the order of dilation is switched.

Answer to Problem 15E

No, the final image will not change if the order of dilation is switched.

Explanation of Solution

Given information:

The vertices of the triangle ABC given are:

  $\text{A}\left(-2,4\right),\text{B}\left(4,0\right)\text{ and C}\left(2,-4\right)$

On dilating the triangle using a scale factor of $0.25$ and then dilating the image using a scale factor of $2$ , the coordinates are $\text{A''}\left(-1,2\right),\text{B''}\left(2,0\right)\text{ and C''}\left(1,-2\right)$ .

Calculation :

Due to the commutative property of multiplication, the order of multiplication does not affect the result, so $2\times 0.25=0.5=0.25\times 2$ .

Hence, the final image will not change if the order of dilation is switched