Find the dilation of 1/2 for the given shape centered at the origin.

Question

a=(2,2)
b=(0,8)
c=(4,-6)

Answer 1

To find the dilation of a shape centered at the origin with a scale factor of $\frac{1}{2}$, you multiply each coordinate of the points by the scale factor.

Given the points:

• $a=(2,2)$
• $b=(0,8)$
• $c=(4,-6)$

We apply the dilation transformation $(x,y)\to (\frac{1}{2}x,\frac{1}{2}y)$.

1. For point $a=(2,2)$: $$a^{\prime }=(\frac{1}{2}\times 2,\frac{1}{2}\times 2)=(1,1)$$

2. For point $b=(0,8)$: $$b^{\prime }=(\frac{1}{2}\times 0,\frac{1}{2}\times 8)=(0,4)$$

3. For point $c=(4,-6)$: $$c^{\prime }=(\frac{1}{2}\times 4,\frac{1}{2}\times -6)=(2,-3)$$

Thus, the dilated points are:

• $a^{\prime }=(1,1)$
• $b^{\prime }=(0,4)$
• $c^{\prime }=(2,-3)$

In summary, after performing the dilation by a factor of $\frac{1}{2}$ centered at the origin, the new coordinates of the points are:

• $a^{\prime }=(1,1)$
• $b^{\prime }=(0,4)$
• $c^{\prime }=(2,-3)$