banner

Figure ABCD is a Trapezoid with point A (0, −4). What rule would rotate the figure 270° clockwise?

This question aims to find the type of rule that would be applied to the trapezoid ABCD with a point A( 0, -4 ) to rotate it to 270° in the clockwise direction.

A quadrilateral having two sides parallel to each other is called a trapezoid. This four-sided figure is also called a trapezium. When we need to find the rotation of a point in the trapezoid, we use the rotation matrix. A transformation matrix rotated in such a way that all its elements get rotated in Euclidean space then it is called a rotation matrix.

The order of the rotation matrix is $ n \times n $ in the n-dimensional space. Similarly, a matrix in a 3-D space will have an order of $ 3 \times 3 $.

Expert Answer

The rotation of a point ( x, y ) in the clockwise direction along an angle $ \theta $ in the coordinate plane is given by the rotation matrix. The order of the rotation matrix is $ n \times n $ in the n-dimensional space.

\begin{bmatrix}
\cos \theta & \sin \theta \\
– \sin \theta & \cos \theta
\end{bmatrix}

By putting the value of the angle $ \theta = 270 ° $

\begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}

The rotation of matrix rule is applied as:

\[ \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}   \begin{bmatrix}
0 & 4
\end{bmatrix}  \]

By multiplying the matrix with 0 and 4:

\[  \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
0  \cos 270  + 4  \sin 270 \\
– 0  \sin 270  + 4 \cos 270
\end{bmatrix}  \]

\[  \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
4  \ sin  270 \\
4  \ cos 270
\end{bmatrix}  \]

Numerical Results

The rule to find the rotation of a trapezoid in the clockwise 270 ° is rotation rule that is given by:

$  \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
4  \ sin  270 \\
4  \ cos 270
\end{bmatrix} $

Example

Rotate the trapezoid having a point ( 0, -3) in the clockwise direction along the angle $ \theta $.

\begin{bmatrix}
\cos \theta & \sin \theta \\
– \sin \theta & \cos \theta
\end{bmatrix}

By putting the value of the angle $ \theta = 270 ° $

\begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}

The rotation of matrix rule is applied as:

\[ \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
\cos 270 & \sin 270 \\
– \sin 270 & \cos 270
\end{bmatrix}   \begin{bmatrix}
0 & 3
\end{bmatrix}  \]

By multiplying the matrix with 0 and 3:

\[  \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
0  \cos 270  + 3  \sin 270 \\
– 0  \sin 270  + 3 \cos 270
\end{bmatrix}  \]

\[  \begin{bmatrix}
x \\
y
\end{bmatrix}     = \begin{bmatrix}
3  \ sin  270 \\
3  \ cos 270
\end{bmatrix}  \]

Image/Mathematical drawings are created in Geogebra.

5/5 - (11 votes)