Given triangle ACD and triangle ABE, what is AB? What is the scale factor of the dilation shown? Is Jaquan correct, and why?

Steps to Solve

1. Determine the sides of triangle ABE

In triangle ABE, we need to find the length of segment $AB$. We have the lengths of sides in triangle ACD: $AC = 12$ and $AD = 15$. Using the triangle similarity properties, we will set up a proportion.

2. Set up the proportion using similar triangles

Since triangles ACD and ABE are similar, we can set up the proportion as:

$$ \frac{AB}{AC} = \frac{BE}{AD} $$

Substituting known values:

$$ \frac{AB}{12} = \frac{9}{15} $$

3. Solve for AB

To find $AB$, we will cross-multiply and solve:

$$ AB \cdot 15 = 12 \cdot 9 $$

Calculating the right side:

$$ AB \cdot 15 = 108 $$

Now, divide both sides by 15:

$$ AB = \frac{108}{15} $$

Simplifying gives:

$$ AB = 7.2 $$

4. Finding the scale factor for the dilation

For triangles GDF and G'F'E', find the scale factor by examining the ratio of corresponding sides. For example, using side $GD$ and $G'F'$:

Scale factor $k$ is given by:

$$ k = \frac{GD}{G'F'} = \frac{8}{4} = 2 $$

5. Assess triangle similarity based on angles

For Jaquan's statement, one triangle has angles measuring $26^\circ$ and $35^\circ$, while the other has angles measuring $119^\circ$ and $26^\circ$.

Since the sum of angles in a triangle is $180^\circ$, calculate the third angle for each:

For the first triangle:

$$ 180 - 26 - 35 = 119^\circ $$

Both triangles now have angles of $26^\circ$ and $119^\circ$, enabling us to conclude that both triangles are similar by the AA postulate.