In the image, triangle ABC is dilated with reference to the origin to create triangle A'B'C'. What is the magnitude of dilation?

Question

(1 point)
Triangle ABC is dilated using a magnitude of
, which results in triangle A'B'C'.

Answer 1

To determine the magnitude of the dilation from triangle ABC to triangle A'B'C', we need to know the coordinates of the vertices of both triangles. The magnitude of dilation \( k \) can be found using the formula:

\[ k = \frac{\text{Distance from the origin to a point in triangle A'B'C'}}{\text{Distance from the origin to the corresponding point in triangle ABC}} \]

If you provide the coordinates of the vertices of triangles ABC and A'B'C', I can help you calculate the magnitude of the dilation. Please share the coordinates for further assistance!

Answer 2

A = (4,2) B = (-2,6) C = (-6,-4)
A' = (2,1) B' = (-1,3) C' = (-6,4)

Answer 3

To find the magnitude of dilation \( k \) from triangle ABC to triangle A'B'C', we can calculate the distances from the origin to each of the corresponding points in both triangles and then use the formula for dilation.

1. Calculate the distances from the origin to each point in triangle ABC:

   • \( A = (4, 2) \): \[ d_A = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

   • \( B = (-2, 6) \): \[ d_B = \sqrt{(-2)^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \]

   • \( C = (-6, -4) \): \[ d_C = \sqrt{(-6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

2. Calculate the distances from the origin to each point in triangle A'B'C':

   • \( A' = (2, 1) \): \[ d_{A'} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]

   • \( B' = (-1, 3) \): \[ d_{B'} = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

   • \( C' = (-6, 4) \): \[ d_{C'} = \sqrt{(-6)^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13} \]

3. Determine the dilation magnitude \( k \) for each pair of corresponding points:

   • For points A and A': \[ k = \frac{d_{A'}}{d_A} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2} \]

   • For points B and B': \[ k = \frac{d_{B'}}{d_B} = \frac{\sqrt{10}}{2\sqrt{10}} = \frac{1}{2} \]

   • For points C and C': \[ k = \frac{d_{C'}}{d_C} = \frac{2\sqrt{13}}{2\sqrt{13}} = 1 \]

4. Conclusion: From the calculated values, it appears that the dilation is consistent for points A and B, where the magnitude of dilation is \( k = \frac{1}{2} \). However, for point C, the dilation is not consistent (there's no dilation).

Therefore, the magnitude of dilation from triangle ABC to triangle A'B'C' is \( \frac{1}{2} \).