Question

graph the image of $\triangle uvw$ after a dilation with a scale factor of $\frac{1}{4}$, centered at the origin.

Explanation:

Step1: Recall dilation rule

For a dilation centered at the origin with scale - factor $k$, if a point $(x,y)$ is dilated, the new point $(x',y')$ is given by $(x',y')=(k x,k y)$. Here $k = \frac{1}{4}$.

Step2: Identify vertices of $\triangle UVW$

Assume the coordinates of the vertices of $\triangle UVW$ are $U(x_U,y_U)$, $V(x_V,y_V)$, and $W(x_W,y_W)$. From the graph, if we assume $U(8, - 4)$, $V(8,0)$, and $W(-8,-8)$.

Step3: Calculate new coordinates

For point $U$:
$x_U'=\frac{1}{4}\times8 = 2$ and $y_U'=\frac{1}{4}\times(-4)=-1$.
For point $V$:
$x_V'=\frac{1}{4}\times8 = 2$ and $y_V'=\frac{1}{4}\times0 = 0$.
For point $W$:
$x_W'=\frac{1}{4}\times(-8)=-2$ and $y_W'=\frac{1}{4}\times(-8)=-2$.

Step4: Graph new triangle

Plot the points $U'(2,-1)$, $V'(2,0)$, and $W'(-2,-2)$ on the same coordinate - grid and connect them to form the dilated $\triangle U'V'W'$.

Answer:

Graph the triangle with vertices $(2,-1)$, $(2,0)$, and $(-2,-2)$.