To solve for $x$ and $y$ in the diagram using the Pythagorean Theorem and ratios:

Observations:

1. Two right triangles are present:
   • Smaller triangle $\triangle AEF$
   • Larger triangle $\triangle ABC$
2. $\triangle AEF \sim \triangle ABC$ (similar triangles due to AA similarity).

Known values:

• $AE = 5$, $AF = 13$
• $CE = 10$, and $AB = y$.

Step-by-Step Solution:

1. Solve $x$ (EF):

For $\triangle AEF$, apply the Pythagorean Theorem: $AF^2 = AE^2 + EF^2$ Substitute the known values: $13^2 = 5^2 + x^2$ $169 = 25 + x^2$ $x^2 = 144 \implies x = 12$

2. Solve $y$ (AB):

Since $\triangle AEF \sim \triangle ABC$, the sides are proportional: $\frac{AE}{AC} = \frac{AF}{AB}$ From the diagram:

• $AC = AE + CE = 5 + 10 = 15$.

Substitute values: $\frac{5}{15} = \frac{13}{y}$ Simplify the ratio: $\frac{1}{3} = \frac{13}{y}$ Cross multiply: $y = 13 \times 3 = 39$

Final Results:

• $x = 12$
• $y = 39$

Would you like further explanation or clarification on any step?

Related Questions:

1. How can triangle similarity help in solving proportions?
2. What are the key properties of similar triangles?
3. How does the Pythagorean Theorem apply in 3D problems?
4. Can the values of $x$ and $y$ be checked using trigonometry?
5. How does scaling work in proportional triangles?

Tip:

Always verify results when using multiple theorems by plugging them back into the relationships to check for consistency!