Use (Triangle) PQR with coordinates P(2, –1), Q(0, 5), and R(–2, 0).
Which matrix expression can be used to translate (triangle)PQR 1 unit left and 3 units up?
To translate a figure, we need to add or subtract values to its coordinate points. In this case, to translate triangle PQR 1 unit left and 3 units up, we need to subtract 1 from the x-coordinates and add 3 to the y-coordinates of its vertices.
Let's denote the translation vector as T = (-1, 3). We can express the translation of a point (x, y) using matrix notation as:
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix}
Expanding the matrix equation, we have:
x' = x - 1
y' = y + 3
Now, let's apply this matrix expression to each vertex of triangle PQR:
For vertex P(2, -1):
x' = 2 - 1 = 1
y' = -1 + 3 = 2
So, the translated coordinates for vertex P are P'(1, 2).
For vertex Q(0, 5):
x' = 0 - 1 = -1
y' = 5 + 3 = 8
So, the translated coordinates for vertex Q are Q'(-1, 8).
For vertex R(-2, 0):
x' = -2 - 1 = -3
y' = 0 + 3 = 3
So, the translated coordinates for vertex R are R'(-3, 3).
Therefore, the translated triangle P'Q'R' can be expressed as:
P'(1, 2), Q'(-1, 8), R'(-3, 3).
Hence, the matrix expression to translate triangle PQR 1 unit left and 3 units up is:
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix}