1. **State the problem:** We have two right triangles sharing a vertex, with the larger triangle having hypotenuse $w$ and base $x$, and the smaller triangle inside it having legs $y$ and $z$. The smaller triangle's leg is 15, and the total base length of both triangles combined is 75. We need to find $x$, $y$, $z$, and $w$. 2. **Identify similarity:** Since the triangles share a vertex and both have right angles, the two triangles are similar by AA (Angle-Angle) similarity. 3. **Set up ratios from similarity:** Let the smaller triangle's base be 15 and the larger triangle's base be $x$. The total base is $x + 15 = 75$, so $$x + 15 = 75 \implies x = 60.$$ 4. **Use similarity ratios:** The ratio of corresponding sides in similar triangles is constant. Let the smaller triangle's legs be $y$ and $z$, and the larger triangle's legs be $x=60$ and $w$ (hypotenuse). Since the smaller triangle's base is 15 and the larger's is 60, the scale factor from smaller to larger is $$\frac{60}{15} = 4.$$ 5. **Find $y$ and $z$:** The smaller triangle's leg corresponding to $x$ is 15, so the other leg $y$ corresponds to $z$ in the larger triangle scaled by 4: $$z = 4y.$$ 6. **Use Pythagoras in smaller triangle:** The smaller triangle has legs 15 and $y$, so $$15^2 + y^2 = z^2.$$ 7. **Substitute $z = 4y$ into the equation:** $$15^2 + y^2 = (4y)^2$$ $$225 + y^2 = 16y^2$$ $$225 = 15y^2$$ $$y^2 = \frac{225}{15} = 15$$ $$y = \sqrt{15} \approx 3.87.$$ 8. **Find $z$:** $$z = 4y = 4 \times 3.87 = 15.49.$$ 9. **Find $w$ (hypotenuse of larger triangle):** Use Pythagoras: $$w^2 = x^2 + z^2 = 60^2 + 15.49^2 = 3600 + 240.1 = 3840.1$$ $$w = \sqrt{3840.1} \approx 61.98.$$ **Final answers:** $$x = 60, \quad y = \sqrt{15} \approx 3.87, \quad z = 15.49, \quad w = 61.98.$$