1. **Problem Statement:** We are given two right triangles with smaller right triangles inside them. We need to find relationships between the sides using the Pythagorean theorem. --- 2. **Recall the Pythagorean theorem:** For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the relationship is: $$a^2 + b^2 = c^2$$ This applies to both the smaller and larger triangles. --- 3. **First triangle (top-left):** - Smaller triangle sides: legs $12$ and $y$, hypotenuse $x$. - Larger triangle sides: one leg $z$, hypotenuse $x + 7$. Apply Pythagorean theorem to the smaller triangle: $$12^2 + y^2 = x^2$$ which simplifies to: $$144 + y^2 = x^2$$ Apply Pythagorean theorem to the larger triangle: $$z^2 + (x + 7)^2 = \text{(other leg)}^2$$ But since the other leg is not given, we focus on the smaller triangle for now. --- 4. **Second triangle (center):** - Smaller triangle legs: $9$ and $z$. - Larger triangle legs: $x$ and $y$. - Larger triangle hypotenuse: $15$. Apply Pythagorean theorem to the larger triangle: $$x^2 + y^2 = 15^2 = 225$$ Apply Pythagorean theorem to the smaller triangle: $$9^2 + z^2 = \text{(hypotenuse of smaller triangle)}^2$$ The hypotenuse of the smaller triangle is not given explicitly, so we keep it as $h$. --- 5. **Use the relationships from the first triangle:** From step 3, we have: $$x^2 = 144 + y^2$$ From step 4, we have: $$x^2 + y^2 = 225$$ Substitute $x^2$ from the first into the second: $$144 + y^2 + y^2 = 225$$ Simplify: $$144 + 2y^2 = 225$$ Subtract 144 from both sides: $$2y^2 = 225 - 144 = 81$$ Divide both sides by 2: $$\cancel{2}y^2 = \frac{81}{\cancel{2}}$$ $$y^2 = \frac{81}{2}$$ Take the square root: $$y = \sqrt{\frac{81}{2}} = \frac{9}{\sqrt{2}} = \frac{9\sqrt{2}}{2}$$ --- 6. **Find $x$ using $y$:** Recall: $$x^2 = 144 + y^2 = 144 + \frac{81}{2} = \frac{288}{2} + \frac{81}{2} = \frac{369}{2}$$ So: $$x = \sqrt{\frac{369}{2}} = \frac{\sqrt{369}}{\sqrt{2}} = \frac{3\sqrt{41}}{\sqrt{2}} = \frac{3\sqrt{82}}{2}$$ --- 7. **Summary of results:** - $y = \frac{9\sqrt{2}}{2}$ - $x = \frac{3\sqrt{82}}{2}$ These values satisfy the Pythagorean relationships in the given triangles. --- **Final answers:** $$y = \frac{9\sqrt{2}}{2}, \quad x = \frac{3\sqrt{82}}{2}$$