Question

If triangle ABC is transformed into A'B'C' what kind of transformation is
A=(0,5) B=(5,7) C=(7,0) and A'=(0,2.5) B'=(2.5,3.5) C'=(3.5,0)

Answer 1

To determine the type of transformation from triangle ABC to triangle A'B'C', let's evaluate the coordinates of the points provided:

Original triangle (ABC):
- A = (0, 5)
- B = (5, 7)
- C = (7, 0)

Transformed triangle (A'B'C'):
- A' = (0, 2.5)
- B' = (2.5, 3.5)
- C' = (3.5, 0)

We can analyze the transformation by comparing the coordinates of the original triangle with those of the transformed triangle.

1. **Check Scale Factor**:
- The coordinates of A' can be derived from A by multiplying the y-coordinate by a scale factor. Specifically, for A:
\[
A' = (0, 5 \times 0.5) = (0, 2.5)
\]
- For B:
\[
B' = (5 \times 0.5, 7 \times 0.5) = (2.5, 3.5)
\]
- For C:
\[
C' = (7 \times 0.5, 0 \times 0.5) = (3.5, 0)
\]

2. **Determine the Transformation**:
From the above calculations, we can see that each point is scaled down by a factor of \(0.5\) in both the x and y coordinates. Therefore, the transformation can be described as a dilation centered at the origin with a scale factor of \(0.5\).

Thus, the transformation that takes triangle ABC to triangle A'B'C' is a **dilation** with a scale factor of \(0.5\) centered at the origin.