1. **Stating the problem:** We have a right triangle with the top horizontal side split into two segments of lengths 20 and 15, and a perpendicular segment $x$ dropping from the split point to the bottom vertex, which forms a right angle. 2. **Understanding the setup:** The triangle is split into two smaller right triangles by the perpendicular segment $x$. The two segments on the top side are 20 and 15. 3. **Using the Pythagorean theorem:** Let the height of the triangle be $x$. The two smaller right triangles share this height $x$ and have bases 20 and 15 respectively. 4. **Expressing the hypotenuses:** Let the hypotenuses of the two smaller triangles be $h_1$ and $h_2$. Then: $$h_1 = \sqrt{20^2 + x^2} = \sqrt{400 + x^2}$$ $$h_2 = \sqrt{15^2 + x^2} = \sqrt{225 + x^2}$$ 5. **Using the right angle at the bottom vertex:** The two hypotenuses $h_1$ and $h_2$ form the sides of the larger triangle meeting at the right angle. Therefore, by the Pythagorean theorem: $$h_1^2 + h_2^2 = (20 + 15)^2 = 35^2 = 1225$$ 6. **Substitute the expressions for $h_1$ and $h_2$:** $$ (400 + x^2) + (225 + x^2) = 1225 $$ 7. **Simplify:** $$ 400 + 225 + 2x^2 = 1225 $$ $$ 625 + 2x^2 = 1225 $$ 8. **Isolate $x^2$:** $$ 2x^2 = 1225 - 625 $$ $$ 2x^2 = 600 $$ 9. **Divide both sides by 2:** $$ x^2 = \cancel{\frac{600}{2}}^{300} $$ 10. **Take the square root:** $$ x = \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} $$ **Final answer:** $$ x = 10\sqrt{3} $$