Chapter 12, Problem 11CST

a

To determine

To find: The coordinates of the vertices.

Answer to Problem 11CST

the point $\text{<msup><mo>A</mo><mo>′</mo></msup> (-2,3), <msup><mo>B</mo><mo>′</mo></msup> (-3,4), and <msup><mo>C</mo><mo>′</mo></msup> (-6,2)}$ is vertices of the $\Delta \text{<msup><mo>A</mo><mo>′</mo></msup><msup><mo>B</mo><mo>′</mo></msup><msup><mo>C</mo><mo>′</mo></msup>}\text{.}$

Explanation of Solution

Given information: $$The vertices of the image of $\Delta \text{ABC}$ after a reflection in $\text{y-axis}$ and vertices is $\text{A(3,2), B(3,4) and C(6,2) of the}$

  $\Delta \text{ABC}$ .

Calculation: The reflection of the point $(\text{x,y)}$ across $\text{y-axis}$ is the point $\text{(-x,y)}\text{.}$

From the given information, the vertices $\text{A(3,2), B(3,4) and C(6,2)}$ of the $\Delta \text{ABC}$ reflect

across $\text{y-axis}$ is the point of $\text{<msup><mo>A</mo><mo>′</mo></msup> (-2,3), <msup><mo>B</mo><mo>′</mo></msup> (-3,4), and <msup><mo>C</mo><mo>′</mo></msup> (-6,2)}$ .

hence, the point $\text{<msup><mo>A</mo><mo>′</mo></msup> (-2,3), <msup><mo>B</mo><mo>′</mo></msup> (-3,4), and <msup><mo>C</mo><mo>′</mo></msup> (-6,2)}$ is vertices of the $\Delta \text{<msup><mo>A</mo><mo>′</mo></msup><msup><mo>B</mo><mo>′</mo></msup><msup><mo>C</mo><mo>′</mo></msup>}\text{.}$

  

From the image we let, $\text{<msup><mo>A</mo><mo>′</mo></msup> =D, <msup><mo>B</mo><mo>′</mo></msup> =E, and <msup><mo>C</mo><mo>′</mo></msup> =F}$ .

b

To determine

To find: The coordinates of the vertices.

Answer to Problem 11CST

the point $\text{<msup><mo>A</mo><mo>″</mo></msup> (2,-3), <msup><mo>B</mo><mo>″</mo></msup> (3,-4), and <msup><mo>C</mo><mo>″</mo></msup> (6,-2)}$ is vertices of the $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$

Explanation of Solution

Given information: The vertices of the image of $\Delta \text{<msup><mo>A</mo><mo>′</mo></msup><msup><mo>B</mo><mo>′</mo></msup><msup><mo>C</mo><mo>′</mo></msup>}$ after a ${180}^{\circ }$ rotation about the origin and $\text{<msup><mo>A</mo><mo>′</mo></msup> (-2,3), <msup><mo>B</mo><mo>′</mo></msup> (-3,4), and <msup><mo>C</mo><mo>′</mo></msup> (-6,2)}$ .

Calculation: The rotation ${180}^{\circ }$ about the origin of the point $(\text{x,y)}$ across the point is $\text{(-x,-y)}$ .

From the given information, the vertices $\text{<msup><mo>A</mo><mo>′</mo></msup> (-2,3), <msup><mo>B</mo><mo>′</mo></msup> (-3,4), and <msup><mo>C</mo><mo>′</mo></msup> (-6,2)}$ of the $\Delta \text{<msup><mo>A</mo><mo>′</mo></msup><msup><mo>B</mo><mo>′</mo></msup><msup><mo>C</mo><mo>′</mo></msup>}$ rotate ${180}^{\circ }$ about the origin is the point of $\text{<msup><mo>A</mo><mo>″</mo></msup> (2,-3), <msup><mo>B</mo><mo>″</mo></msup> (3,-4), and <msup><mo>C</mo><mo>″</mo></msup> (6,-2)}$ .

Hence, the point $\text{<msup><mo>A</mo><mo>″</mo></msup> (2,-3), <msup><mo>B</mo><mo>″</mo></msup> (3,-4), and <msup><mo>C</mo><mo>″</mo></msup> (6,-2)}$ is vertices of the $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$ .

  

From the image, we let $\text{<msup><mo>A</mo><mo>″</mo></msup> =D, <msup><mo>B</mo><mo>″</mo></msup> =E, and <msup><mo>C</mo><mo>″</mo></msup> =F}$

c

To determine

To move: Using one transformation.

Answer to Problem 11CST

The reflection of the vertices of $\Delta \text{ABC}$ across the $\text{x-axis}$ then we get $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$ .

Explanation of Solution

Given information: Move $\Delta \text{ABC}$ to $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$ using one transformation.

Description: Using one transformation to move $\Delta \text{ABC}$ to $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$ .If we reflect the vertices of $\Delta \text{ABC}$ across the $\text{x-axis}$ is the point $\text{(x,-y)}$ . Then we get $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$

Hence , the reflection of the vertices of $\Delta \text{ABC}$ across the $\text{x-axis}$ then we get $\Delta \text{<msup><mo>A</mo><mo>″</mo></msup><msup><mo>B</mo><mo>″</mo></msup><msup><mo>C</mo><mo>″</mo></msup>}$ .