A (-1,-1)

Question

B (-1,1)
C (2,3)
D (2,-1)

A' (-2,-2)
B' (-2,2)
C' (4,6)
D' (4,-2)

In the image, polgon ABCD is dilated to form polygon A' B' C' D' with respect to the origin. What is the magnitude of dilation?

a. 1
b. 3
c. 2
d. 1/2

Answer 1

To find the magnitude of dilation, we can use the formula for dilation about the origin. The formula states that the coordinates of a point $P(x,y)$ after dilation by a factor of $k$ will be $P^{\prime }(kx,ky)$.

We can find the dilation factor $k$ by comparing the coordinates of corresponding points in the original polygon and the dilated polygon. Let's take one pair of corresponding points. Using points $A$ and $A^{\prime }$:

• Original point $A$: $(-1,-1)$
• Dilated point $A^{\prime }$: $(-2,-2)$

Now we calculate the dilation factor $k$:

$$A^{\prime }=k\cdot A$$ $$(-2,-2)=k\cdot (-1,-1)$$

This gives us the equations:

• $-2=k\cdot (-1)$
• $-2=k\cdot (-1)$

From either equation, we can solve for $k$:

$$k=\frac{-2}{-1}=2$$

Now we can confirm this dilation factor with another pair of corresponding points, say points $B$ and $B^{\prime }$:

• Original point $B$: $(-1,1)$
• Dilated point $B^{\prime }$: $(-2,2)$

Calculating $k$ again:

$$(-2,2)=k\cdot (-1,1)$$

This gives us:

• $-2=k\cdot (-1)$ leads to $k=2$
• $2=k\cdot 1$ also leads to $k=2$

Since the dilation factor is consistent across multiple points, we conclude that the magnitude of dilation is

c. 2.