Question

Triangle ABC
𝐴
𝐵
𝐶
has vertices point A(2,1)
𝐴
(
2
,
1
)
, point B(4,3)
𝐵
(
4
,
3
)
, and point C(−2,5)
𝐶
(
−
2
,
5
)
. Find the coordinates of A′
𝐴
′
, B′
𝐵
′
, and C′
𝐶
′
after a dilation with a scale factor of 7 and a center point of dilation at the origin (0,0)
(
0
,
0
)
.(2 points)
A′=
𝐴
′
=
(,)
B′=
𝐵
′
=
(,)
C′=
𝐶
′
=
(,)

Answer 1

To perform a dilation of a triangle with a scale factor of \( k \) and a center of dilation at the origin (0, 0), you can use the following formula for the coordinates of the dilated points:

\[
(x', y') = (k \cdot x, k \cdot y)
\]

In this case, the scale factor \( k \) is 7. Now we will apply this formula to each vertex of triangle ABC:

1. **For point A(2, 1)**:
\[
A' = (7 \cdot 2, 7 \cdot 1) = (14, 7)
\]

2. **For point B(4, 3)**:
\[
B' = (7 \cdot 4, 7 \cdot 3) = (28, 21)
\]

3. **For point C(−2, 5)**:
\[
C' = (7 \cdot (-2), 7 \cdot 5) = (-14, 35)
\]

Thus, the coordinates after the dilation are:

- \( A' = (14, 7) \)
- \( B' = (28, 21) \)
- \( C' = (-14, 35) \)

So, the final answers are:
- \( A' = (14, 7) \)
- \( B' = (28, 21) \)
- \( C' = (-14, 35) \)