1. **Problem Statement:** We have a trapezoid with the top base $11$ cm, the bottom base $26$ cm, and one non-parallel side (left side) $17$ cm. We need to find the height $h$ of the trapezoid. 2. **Formula and Explanation:** The height $h$ is the perpendicular distance between the two parallel bases. We can use the Pythagorean theorem by dropping a perpendicular from the top base to the bottom base, creating a right triangle. 3. **Step-by-step solution:** - Let the trapezoid be $ABCD$ with $AB=11$ cm (top base), $CD=26$ cm (bottom base), and $AD=17$ cm (left side). - Drop a perpendicular from $B$ to $CD$ at point $E$, so $BE = h$ (height), and $DE = x$ (segment on bottom base). - Since $CD = 26$ cm and $AB = 11$ cm, the difference is $26 - 11 = 15$ cm. - The segment $CE$ is then $15 - x$ cm. 4. **Using the right triangle $ADE$:** - $AD = 17$ cm is the hypotenuse. - $DE = x$ is the base. - $AE = h$ is the height. 5. **Apply the Pythagorean theorem:** $$17^2 = h^2 + x^2$$ 6. **Using the trapezoid properties:** - The top base $AB$ is parallel to $CD$. - The segment $BE = h$ is perpendicular to $CD$. - The length $AE$ is $11$ cm (top base) plus $x$ (segment on bottom base), so $AE = 11 + x$. 7. **Since $AE$ lies along the bottom base, and $CE = 15 - x$, the total bottom base is $26$ cm:** - So $AE + CE = 11 + x + 15 - x = 26$ cm, which is consistent. 8. **We need to find $h$ and $x$. Using the right triangle $BCE$:** - $BE = h$ (height), - $CE = 15 - x$ (base), - $BC$ is unknown but not needed here. 9. **Using the right triangle $ADE$:** - $AD = 17$ cm, - $DE = x$, - $AE = h$. 10. **From the Pythagorean theorem:** $$h^2 = 17^2 - x^2 = 289 - x^2$$ 11. **From the trapezoid, the horizontal distance between the bases is $15$ cm, so the horizontal leg of the right triangle is $x$, and the other part is $15 - x$. The height $h$ is the same for both triangles.** 12. **Using the right triangle $BCE$ (with base $15 - x$ and height $h$), the length $BC$ is unknown but not needed. We can use the fact that the top base is $11$ cm, so the horizontal distance from $A$ to $E$ is $x + 11$ cm.** 13. **Since $AE$ is along the bottom base, and $DE = x$, the height $h$ is perpendicular to the bases.** 14. **We can solve for $h$ by expressing $h$ in terms of $x$ and then finding $x$ using the trapezoid properties.** 15. **Using the Pythagorean theorem on triangle $ADE$:** $$h = \sqrt{289 - x^2}$$ 16. **Using the Pythagorean theorem on triangle $BCE$ (right triangle with base $15 - x$ and height $h$), the length $BC$ is unknown but not needed.** 17. **Since the trapezoid is not isosceles, we use the fact that the height is the same for both triangles, so:** $$h = \sqrt{289 - x^2}$$ 18. **We can find $x$ by using the fact that the top base is $11$ cm and the bottom base is $26$ cm, so the difference is $15$ cm, which is split into $x$ and $15 - x$.** 19. **Using the Pythagorean theorem on triangle $BCE$ with base $15 - x$ and height $h$, and side $BC$ unknown, we don't have enough information to solve for $x$ directly.** 20. **However, since the trapezoid has one non-parallel side $AD=17$ cm, and the height $h$ is perpendicular, we can use the formula for the height of a trapezoid given the sides:** $$h = \sqrt{AD^2 - \left(\frac{(CD - AB)^2 + AD^2 - BC^2}{2(CD - AB)}\right)^2}$$ 21. **Since $BC$ is not given, we assume the trapezoid is right-angled at $B$, so $BC$ is perpendicular to $AB$, and the height is $h = 17$ cm.** 22. **Therefore, the height $h$ is $17$ cm.** **Final answer:** $$h = 17 \text{ cm}$$