Notice that the dilation is an enlargement. Pick a general triangle so that you can easily find its area. Dilate it with a general scale factor at any center, and find its area. From this, you can find the scale factor.
Example Solution: center at (0,0) and scale factor of 2 Example Graph:
Practice makes perfect
Since the image of the triangle must have a bigger area, we conclude that the dilation must be an enlargement. That is, k> 1. For simplicity, let's consider a right triangle.
Next, let's perform a dilation with its center at the origin. Remember that it must be an enlargement.
Let's simplify the expression for the second area.
Remember that we are also told that the area of the second triangle is four times the area of the first one. Then, we can set the equations below.
A_2 = 4 A_1
A_2 = k^2 * A_1
⇒
k^2 = 4
From the above, we conclude that k=2. Notice that the center of dilation can be any point; here we used the origin for simplicity.
Particular Example
As a particular example, below we draw a triangle ABC and its image under a dilation centered at the origin and scale factor 2.
Keep in mind that this is just a sample triangle and so, your answer may vary.
