riangle ABC

Question

has vertices point A(2, 1)
, point B(−2, 2)
, and point C(−3,−1)
. After dilation, triangle A′B′C′
has vertices point A′(4, 2)
, point B′(−4, 4)
, and point C′(−6,−2)
. Determine the scale factor for the dilation of triangle ABC
with a center point of dilation at the origin (0, 0 )
.(1 point)
The scale factor is
.

Answer 1

To determine the scale factor of the dilation from triangle ABC to triangle A'B'C', we can use the coordinates of the vertices before and after dilation.

The formula for the dilated coordinates from point \( (x, y) \) to \( (kx, ky) \), where \( k \) is the scale factor, can help us determine \( k \).

Let's use point A as an example:

• Original Point A: \( (2, 1) \)
• Dilated Point A': \( (4, 2) \)

We set up the equations based on the dilation formula: \[ k \cdot (2, 1) = (4, 2) \]

This gives us two equations:

1. \( k \cdot 2 = 4 \)
2. \( k \cdot 1 = 2 \)

From the first equation \( k \cdot 2 = 4 \): \[ k = \frac{4}{2} = 2 \]

From the second equation \( k \cdot 1 = 2 \): \[ k = \frac{2}{1} = 2 \]

Both equations yield the same scale factor.

Now, let's verify with the other points.

For point B:

• Original Point B: \( (-2, 2) \)
• Dilated Point B': \( (-4, 4) \)

Using the scale factor \( k = 2 \): \[ k \cdot (-2) = -4 \quad \text{and} \quad k \cdot 2 = 4 \] which both check out since \( 2 \cdot (-2) = -4 \) and \( 2 \cdot 2 = 4 \).

For point C:

• Original Point C: \( (-3, -1) \)
• Dilated Point C': \( (-6, -2) \)

Using the scale factor \( k = 2 \): \[ k \cdot (-3) = -6 \quad \text{and} \quad k \cdot (-1) = -2 \] which are also correct since \( 2 \cdot (-3) = -6 \) and \( 2 \cdot (-1) = -2 \).

Thus, the consistent scale factor from triangle ABC to triangle A'B'C' is:

\[ \text{The scale factor is } 2. \]