Mathematics, 20.09.2019 12:30 arijade1391

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)

Triangle abc is shown on the graph. what are the coordinates of the image of point b after the trian

Answers: 2

Show answers

Answers

Answer from: rina62

(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 moreThe similar problem linkDraw a line K that passes through (3, 5) but not parallel to either x-axis or y-axis linkThe composite function link

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 abc is shown on the graph. what are the coordinates of the image of point b after the trian

Answer from: idontknow113

The coordinate of the vertices of point 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 is $\left( { - 5,3} \right)$ , point is $\left( { - 2,4} \right)$ and point 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 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? link

2. Which undefined term is needed to define an angle? link

3. Look at the figure, which trigonometric ratio should you use to find x? link

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.

Answer from: Quest

see below

step-by-step explanation:

7. there are 10 data points in the set. the middle point is between 5 and 6. the middle point in the top half is 8 and the middle point in the bottom half is 3. to find the iqr , take point 8 and subtract point 3

mid term point 8 = 100 point 3 = 88

iqr = 100-88 = 12

final point 8 = 93 point 3 = 78

iqr = 93 - 78 = 15

the final has the greatest iqr

6. b - there is a high data value that causes the data set to be asymmetrical for the males: the data for males are high and asymmetrical . there is an outlier for the males that skews the data

5. the iqr is a better measure of spread for the movies and basketball games. since the basketball game has an outlier, we do not want to use standard deviation for measurement.

3. the exam median is much higher than the class median. the median line is farther to the right on the exams

2. there is no way to tell what the means are from the box and whisker plots.

Answer from: Quest

ghjmutmum

step-by-step explanation: