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?  A = (4,2) B = (-2,6) C = (-6, -4). A' = (2,1) B' = (-1,3) C' = (-3,-2) (1 point) Triangle ABC is dilated using a magnitude of  , which results in triangle A'B'C'. 

Answer 1

To find the magnitude of dilation, we can use the formula for dilation with respect to the origin, which is defined as:

\[
A' = k \cdot A
\]

where \(k\) is the dilation factor and \(A\) and \(A'\) are the coordinates of the points before and after dilation, respectively.

We can calculate the dilation factor \(k\) by taking a corresponding point from triangle ABC and triangle A'B'C'. Let's take point A as an example:

1. The coordinates of A are (4, 2) and the coordinates of A' are (2, 1).
2. We can find \(k\) by examining the relationship between the coordinates of A and A':

\[
k = \frac{A'_x}{A_x} = \frac{2}{4} = \frac{1}{2}
\]

\[
k = \frac{A'_y}{A_y} = \frac{1}{2} = \frac{1}{2}
\]

Since both the x and y coordinates give us the same dilation factor, we can confirm that the magnitude of dilation is consistent.

Now, let's apply the same calculations to another point, for instance point B:

1. The coordinates of B are (-2, 6) and the coordinates of B' are (-1, 3).

Calculating \(k\) for B:

\[
k = \frac{B'_x}{B_x} = \frac{-1}{-2} = \frac{1}{2}
\]

\[
k = \frac{B'_y}{B_y} = \frac{3}{6} = \frac{1}{2}
\]

This also confirms that \(k = \frac{1}{2}\).

Finally, we can check point C:

1. The coordinates of C are (-6, -4) and the coordinates of C' are (-3, -2).

Calculating \(k\) for C:

\[
k = \frac{C'_x}{C_x} = \frac{-3}{-6} = \frac{1}{2}
\]

\[
k = \frac{C'_y}{C_y} = \frac{-2}{-4} = \frac{1}{2}
\]

Once again, both the calculations yield \(k = \frac{1}{2}\).

Therefore, the magnitude of dilation is:

\[
\boxed{\frac{1}{2}}
\]