Question

a. draw the dilation of triangle abc, with center (0, 0), and scale factor 2. label this triangle abc.
b. draw the dilation of triangle abc, with center (0, 0), and scale factor 1/2. label this triangle abc.
c. is abc a dilation of triangle abc? if yes, what are the center of dilation and the scale factor?

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated with center $(0,0)$ and scale - factor $k$, the new point is $(kx,ky)$.

Step2: Find coordinates for part a

Let the coordinates of $\triangle ABC$ be $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$. For a scale - factor $k = 2$, the coordinates of $\triangle A'B'C'$ are $A'(2x_1,2y_1)$, $B'(2x_2,2y_2)$, $C'(2x_3,2y_3)$. Plot these points and label the triangle.

Step3: Find coordinates for part b

For a scale - factor $k=\frac{1}{2}$, the coordinates of $\triangle A''B''C''$ are $A''(\frac{1}{2}x_1,\frac{1}{2}y_1)$, $B''(\frac{1}{2}x_2,\frac{1}{2}y_2)$, $C''(\frac{1}{2}x_3,\frac{1}{2}y_3)$. Plot these points and label the triangle.

Step4: Analyze part c

To check if $\triangle A''B''C''$ is a dilation of $\triangle A'B'C'$, we use the dilation formula. Let the scale - factor be $k'$ and center of dilation be $(0,0)$. If we go from $A'(2x_1,2y_1)$ to $A''(\frac{1}{2}x_1,\frac{1}{2}y_1)$, we find $k'=\frac{\frac{1}{2}x_1}{2x_1}=\frac{1}{4}$ (similarly for other points). The center of dilation is $(0,0)$.

Answer:

a. To draw $\triangle A'B'C'$: Multiply the coordinates of $A$, $B$, and $C$ by 2 and plot the new points.
b. To draw $\triangle A''B''C''$: Multiply the coordinates of $A$, $B$, and $C$ by $\frac{1}{2}$ and plot the new points.
c. Yes, the center of dilation is $(0,0)$ and the scale factor is $\frac{1}{4}$.