Question

In the image, triangle ABC is dilated with reference to the origin to create triangle A′B′C′ . What is the magnitude of dilation? (2,1) (-1,3) (-3,-2)

Answer 1

To find the magnitude of the dilation that transforms triangle ABC with vertices at A(2, 1), B(-1, 3), and C(-3, -2) to triangle A′B′C′ with reference to the origin, we look at the distances of the points from the origin both before and after the dilation.

The formula for dilation with respect to the origin is given by:
\[
(x', y') = k(x, y)
\]
where \( (x, y) \) are the coordinates of the original points, \( (x', y') \) are the coordinates of the transformed points, and \( k \) is the scale factor of dilation (the magnitude of dilation).

1. **Calculate the distances from the origin:**
For point A(2, 1):
\[
d_A = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}
\]

For point B(-1, 3):
\[
d_B = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]

For point C(-3, -2):
\[
d_C = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

2. **Assuming we need to find the new dilation ratio for points A', B', and C' based on the same frame of reference.**
These new points have different coordinates. For instance, if we define points A′, B′, and C′ as the points after dilation, we’ll need their coordinates to establish the transformation ratio.

If you have specific coordinates for points A', B', and C' defined in terms of a certain scale, please provide those to calculate the exact value of \( k \).

3. **If specific coordinates for the newly dilated triangle A′B′C′ have been provided, you can find \( k \) by comparing distances.**
For example, if new coordinates for A', B', and C' are known, you can compute their distances from the origin and create ratios \( k \) like this:
\[
k = \frac{d_{A'}}{d_A} \quad (and similarly for B and C)
\]

If you have the coordinates of A′B′C′ please share them to proceed with the exact calculation.