Question

Triangle ABC is shown on the graph. What are the coordinates of the image of point B after the triangle is rotated 270° about the origin? (4, 2) (2, 4) (–4, –2) (–2, –4)

Answer 1

(4, 2).

The Answer is A.

Further explanation

For a counterclockwise rotation of 270° about the origin, symbolized by $\boxed{ \ Rot_{270} \ }$, the vertex matrix as a multiplier is

$\left[\begin{array}{ccc}0&1\\-1&0\\\end{array}\right].$

State (x, y) as the initial coordinate and (x', y') as the final coordinate.

The results of rotation are obtained from the multiplication of the matrix with the initial coordinates.  

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}0&1\\-1&0\\\end{array}\right]\left[\begin{array}{ccc}x\\y\\\end{array}\right]$

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}(0)(x) + (1)(y)\\(-1)(x) + (0)(y)\\\end{array}\right]$

$\left[\begin{array}{ccc}x'\\y'\\\end{array}\right] = \left[\begin{array}{ccc}y\\-x\\\end{array}\right]$

It can be concluded that if a point is rotated 270° about the origin, the rule that describes the transformation is $\boxed{ \ (x, y) \rightarrow (y, -x) \ }$

Let's consider the key problem.

Triangle ABC with coordinates of points A (-5, 3), B (-2, 4), and C (-2, 2) is rotated 270° about the origin.

Let's find out the rotation results called A', B', and C' implementing the rule.

$\boxed{ \ (x, y) \rightarrow (y, -x) \ }$

• $\boxed{ \ A(-5, 3) \rightarrow A'(3, 5) \ }$
• $\boxed{ \ B(-2, 4) \rightarrow B'(4, 2) \ }$
• $\boxed{ \ C(-2, 2) \rightarrow C'(2, 2) \ }$

Hence,  the coordinates of the image of point B after the triangle is rotated 270° about the origin is B' (4, 2).

Learn more

1. The similar problem brainly.com/question/2992432
2. Draw a line K that passes through (3, 5) but not parallel to either x-axis or y-axis  brainly.com/question/4691222
3. The composite function brainly.com/question/1691598

Keywords: triangle ABC, is shown on the graph, rotated 270° about the origin, rotation,degrees, transformation geometry, translation, reflection, dilation, multiplier, vertex matrix, initial coordinate, the image, rule, describes

Answer 2

The coordinate of the vertices of point $B$ after the triangle is rotated $270^\circ$ about the origin is $\boxed{\bf{B_1}\left( {4,2} \right)}$ .  

Further explanation:  

The formula for counter clockwise rotation of $\theta$ degree about origin symbolized by $Ro{t_{\left( \theta\right)}}$, the vertex matrix as a multiplier is as follows:  

$Ro{t_{\left( \theta \right)}} =\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta } \\ {\sin \theta }&{\cos \theta } \end{array}} \right]$  

Substitute 270 for $\theta$ in above expression.  

$\begin{aligned} Ro{t_{\left( {270} \right)}} &= \left[ {\begin{array}{*{20}{c}} {\cos 270}&{ - \sin 270} \\ {\sin 270}&{\cos 270} \end{array}} \right]\\ & = \left[ {\begin{array}{*{20}{c}} 0&{ - \left( { - 1} \right)} \\{\left( { - 1} \right)}&0 \end{array}} \right] \\&= \left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right] \\ \end{aligned}$  

Consider $\left( {x,y} \right)$ as initial coordinate and $\left( {{x_1},{y_1}} \right)$ as the final coordinate of the vertices.  

The resultant matrix after rotation of $270^\circ$ is obtained as follows:  $\begin{aligned} \left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{y_1}} \end{array}} \right] &= \left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 1}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] \\ &= \left[ {\begin{array}{*{20}{c}} {0\left( x \right) + 1\left( y \right)} \\ { - 1\left( x \right) + 0\left( y \right)} \end{array}} \right] \\ &= \left[ {\begin{array}{*{20}{c}} y \\ { - x} \end{array}} \right] \\ \end{aligned}$  

From the above expression, it is observed that if a point $\left( {x,y} \right)$ is rotated by $270^\circ$ in counterclockwise direction about the origin, its coordinate becomes$\left( {y, - x} \right)$.  

From the given graph, it is observed that the coordinate of point $A$ is $\left( { - 5,3} \right)$, point $B$ is $\left( { - 2,4} \right)$ and point $C$ is $\left( { - 2,2} \right)$.  

As $\left( {x,y} \right) \to \left( {y, - x} \right)$  

The coordinate of the vertices after rotation ${A_1},{B_1}$ and ${C_1}$ is obtained as follows:  

$\begin{aligned} A\left( { - 5,3} \right) \to {A_1}\left( {3,5} \right) \hfill \\ B\left( { - 2,4} \right) \to {B_1}\left( {4,2} \right) \hfill \\ C\left( { - 2,2} \right) \to {C_1}\left( {2,2} \right) \hfill \\ \end{aligned}$  

Thus, the coordinate of the vertices of point $B$ after the triangle is rotated $270^\circ$ about the origin is $\boxed{\bf{B_1}\left( {4,2} \right)}$.  

The graph of the rotation of triangle ABC is attached below:  

Learn more:  

1. Which rule describes the transformation? brainly.com/question/2992432  

2. Which undefined term is needed to define an angle? brainly.com/question/3717797  

3. Look at the figure, which trigonometric ratio should you use to find x? brainly.com/question/9880052  

Answer Details :  

Grade: Senior School  

Subject: Mathematics  

Chapter: Coordinate geometry.  

Keywords:  

triangle ABC, is shown on the graph, rotated 270° about the origin, rotation, degrees, transformation geometry, translation, reflection, dilation, multiplier, vertex matrix, initial coordinate, the image, rule, describes triangle, similar, similarity, ratio of sides, right triangle, similar triangle, ratio of sides, equal angles, square of hypotenuse, sum, square of legs, sum of square of legs, sum of angle of triangle, property of triangle, triangle ABC.