Question

find the scale and point of dilation that would map cap to cap
scale:________ point of dilation:________
c = (0, 6)
a = (-4, 4)
a = (-3, 2)
c = (-1, 3)
p = (6, 2)
p = (2, 1)

Explanation:

Step1: Recall dilation formula

For a point $(x,y)$ dilated about a center $(a,b)$ by a scale factor $k$, the new - point $(x',y')$ is given by $x'=a + k(x - a)$ and $y'=b + k(y - b)$. Let the center of dilation be $(a,b)$ and the scale factor be $k$. We can use the fact that for a point and its dilated image, we can set up a system of equations. Let's use the points $C(0,6)$ and $C'(-1,3)$.
\[x'=a + k(x - a)\]
\[y'=b + k(y - b)\]
Substituting $x = 0,y = 6,x'=-1,y' = 3$:
\[-1=a + k(0 - a)=a(1 - k)\]
\[3=b + k(6 - b)=b+(6k - bk)\]
Let's also use the points $A(-4,4)$ and $A'(-3,2)$.
\[-3=a + k(-4 - a)=a-4k - ak\]
\[2=b + k(4 - b)=b + 4k - bk\]
Subtract the second - equation for $y$ values from the first - equation for $y$ values:
\[3 - 2=(b + 6k - bk)-(b + 4k - bk)\]
\[1 = 2k\]

Step2: Solve for scale factor $k$

From $1 = 2k$, we get $k=\frac{1}{2}$.

Step3: Solve for center of dilation $(a,b)$

Substitute $k=\frac{1}{2}$ into the equation for the $x$ - coordinate using the points $C$ and $C'$:
\[-1=a+\frac{1}{2}(0 - a)\]
\[-1=a-\frac{1}{2}a=\frac{1}{2}a\]
\[a=-2\]
Substitute $k = \frac{1}{2}$ into the equation for the $y$ - coordinate using the points $C$ and $C'$:
\[3=b+\frac{1}{2}(6 - b)\]
\[3=b + 3-\frac{1}{2}b\]
\[3=3+\frac{1}{2}b\]
\[b = 0\]

Answer:

Scale: $\frac{1}{2}$
Point of Dilation: $(-2,0)$