Question

B (1,7) 5- 4- 3 2- 1- A (5,5) < (1 point) < x + 1 2 5 6 -0+ -10 -1- C (3,2) 3 4 Dilate the triangle using a magnitude of and a center of dilation at (2, 2) to form triangle A' B'C'. What are the coordinates of the dilated triangle? Α΄(8,8), Β΄(0,12), C'(4,2) Ο Α΄(7,7), Β΄(3,9), C'(5,4) Α΄(3.5,3.5), Β' (1.5,4.5), C'(2.5,2) Ο Α΄(2.5.2.5), Β΄(0.5,3.5), C'(1.5,1)

Answer 1

To dilate the triangle with a center of dilation at (2, 2) and a magnitude of 2 (it appears that the magnitude is not explicitly stated but let's assume 2 for this example), we can use the following formula for dilation:

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

where \( (x_c, y_c) \) is the center of dilation, \( k \) is the magnitude of the dilation, and \( (x', y') \) are the new coordinates after dilation.

Given the coordinates of the vertices of triangle ABC:
- A(5, 5)
- B(1, 7)
- C(3, 2)

Let's perform the dilation for each vertex:

### For point A(5, 5):
- \( (x_c, y_c) = (2, 2) \)
- \( k = 2 \)
- New coordinates:
\[
A' = (2 + 2(5 - 2), 2 + 2(5 - 2)) = (2 + 2 \cdot 3, 2 + 2 \cdot 3) = (2 + 6, 2 + 6) = (8, 8)
\]

### For point B(1, 7):
\[
B' = (2 + 2(1 - 2), 2 + 2(7 - 2)) = (2 + 2(-1), 2 + 2(5)) = (2 - 2, 2 + 10) = (0, 12)
\]

### For point C(3, 2):
\[
C' = (2 + 2(3 - 2), 2 + 2(2 - 2)) = (2 + 2(1), 2 + 2(0)) = (2 + 2, 2 + 0) = (4, 2)
\]

Now we have the coordinates of the dilated triangle A' B' C':
- A' (8, 8)
- B' (0, 12)
- C' (4, 2)

Therefore, the coordinates of the dilated triangle are:

**A' (8, 8), B' (0, 12), C' (4, 2)**

So the correct option is: **Α΄(8,8), Β΄(0,12), C'(4,2)**.