Question

plot the image of point $b$ under a dilation about point $p$ with a scale factor of $\frac{1}{4}$.

Explanation:

Step1: Recall dilation formula

If we have a dilation about a center point $P(x_p,y_p)$ with a scale - factor $k$ and a point $B(x_b,y_b)$, the coordinates of the dilated point $B'$ are given by $B'(x_p + k(x_b - x_p),y_p + k(y_b - y_p))$. First, we need to assume coordinates for $P$ and $B$. Let's assume $P(x_p,y_p)=(1,1)$ and $B(x_b,y_b)=(7,7)$ for the sake of illustration (since no coordinates are given in the problem, the general method is shown).

Step2: Calculate $x$ - coordinate of the dilated point

Using the formula $x'=x_p + k(x_b - x_p)$ with $k = \frac{1}{4}$, $x_p = 1$ and $x_b = 7$.
$x'=1+\frac{1}{4}(7 - 1)=1+\frac{1}{4}\times6=1 + \frac{3}{2}=\frac{2 + 3}{2}=\frac{5}{2}=2.5$

Step3: Calculate $y$ - coordinate of the dilated point

Using the formula $y'=y_p + k(y_b - y_p)$ with $k=\frac{1}{4}$, $y_p = 1$ and $y_b = 7$.
$y'=1+\frac{1}{4}(7 - 1)=1+\frac{1}{4}\times6=1+\frac{3}{2}=\frac{5}{2}=2.5$

To plot the point in general:

1. Draw a line segment connecting point $P$ and point $B$.
2. Measure the distance from $P$ to $B$.
3. Divide this distance into 4 equal parts (since the scale - factor is $\frac{1}{4}$).
4. Locate the point on the line segment from $P$ to $B$ that is $\frac{1}{4}$ of the way from $P$ to $B$. This is the dilated point.

Answer:

Plot the point on the line segment connecting $P$ and $B$ that is $\frac{1}{4}$ of the distance from $P$ to $B$. If coordinates are used (assuming $P(1,1)$ and $B(7,7)$), the dilated point has coordinates $(2.5,2.5)$.