Solve the math problems in the image.

Steps to Solve

1. Solve for AD using similar triangles

Notice that triangle $ABC$ and triangle $ADE$ are similar triangles. We can set up the following proportion: $\frac{AB}{AD} = \frac{BC}{DE}$

2. Plug in the known values

We know that $AB = 10$, $BC = 8$, and $DE = 10$. Plugging these values into the proportion: $\frac{10}{AD} = \frac{8}{10}$

3. Solve for AD

Cross-multiply to solve for $AD$: $8 \cdot AD = 10 \cdot 10$ $8 \cdot AD = 100$ $AD = \frac{100}{8} = \frac{25}{2} = 12.5 = 12\frac{1}{2}$

4. Solve for slope of the line

To find the slope of the line, we need to identify two points on the line and use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ From the graph, we can identify two points: $(0, 1)$ and $(2, 0)$.

5. Calculate the slope

Plugging these values into the slope formula: $m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} = -\frac{1}{2}$

6. Solve for EC using similar triangles

Notice that triangle $AED$ and triangle $ABC$ are similar triangles. We are given that $AE=16$, $ED=12$, and $DB=15$. Therefore, $BC = ED + DB = 12 + 15 = 27$. We need to find the length of $EC$. Since triangle $AED$ and triangle $ABC$ are similar triangles, we can write: $\frac{AE}{AB} = \frac{ED}{BC}$ We know $AE = 16$, $ED = 12$, and $BC = 27$. We need to find $AC$ so that we can subtract $AE$ to get $EC$. Notice: $EC = AC - AE$

7. Solve for length of AC

Using the similar triangles $AED$ and $ABC$, we write: $\frac{AE}{AC} = \frac{ED}{BC}$ $\frac{16}{AC} = \frac{12}{27}$ Cross multiplying gives: $12 \cdot AC = 16 \cdot 27$ $12 \cdot AC = 432$ $AC = \frac{432}{12} = 36$

8. Solve for the length of EC

Since we know $AE = 16$ and $AC = 36$, we can find the length of $EC$: $EC = AC - AE = 36 - 16 = 20$