Question

if you dilate A(1,6)B(3,0)C(6,4) to form X'Y'Z' using a magnitude of 2 and a center of dilation t (2,0), what are the coordinates of the dilated triangle

Answer 1

To find the coordinates of the dilated triangle \( X'Y'Z' \) from the triangle \( ABC \) using dilation, we can apply the dilation formula. The formula for dilating a point \( (x, y) \) around a center of dilation \( (h, k) \) with a scale factor \( k \) is:

\[
(x', y') = (h + k(x - h), k + k(y - k))
\]

In this case, the center of dilation \( t \) is \( (2, 0) \), and the scale factor \( k \) is \( 2 \).

### Step 1: Dilation of Point A(1, 6)

Using the formula for point A(1, 6):

\[
x' = 2 + 2(1 - 2) = 2 + 2(-1) = 2 - 2 = 0
\]
\[
y' = 0 + 2(6 - 0) = 0 + 2(6) = 0 + 12 = 12
\]

So, the coordinates of A' are \( (0, 12) \).

### Step 2: Dilation of Point B(3, 0)

Using the formula for point B(3, 0):

\[
x' = 2 + 2(3 - 2) = 2 + 2(1) = 2 + 2 = 4
\]
\[
y' = 0 + 2(0 - 0) = 0 + 0 = 0
\]

So, the coordinates of B' are \( (4, 0) \).

### Step 3: Dilation of Point C(6, 4)

Using the formula for point C(6, 4):

\[
x' = 2 + 2(6 - 2) = 2 + 2(4) = 2 + 8 = 10
\]
\[
y' = 0 + 2(4 - 0) = 0 + 2(4) = 0 + 8 = 8
\]

So, the coordinates of C' are \( (10, 8) \).

### Summary of Dilated Coordinates

The coordinates of the dilated triangle \( X'Y'Z' \) are:
- A' = \( (0, 12) \)
- B' = \( (4, 0) \)
- C' = \( (10, 8) \)

Thus, the final answer is:
- A' = \( (0, 12) \)
- B' = \( (4, 0) \)
- C' = \( (10, 8) \)