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 $3 \times 3$.

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:

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

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

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