Solution: Let the trapezoid have bases of lengths $ a = 120 $ m and $ b = 180 $ m, and equal legs of length $ l = 50 $ m. The height $ h $ can be found by dropping perpendiculars from the ends of the shorter base to the longer base, forming two right triangles on either side and a rectangle in the middle. The difference in the bases is $ b - a = 60 $ m, so each of the right triangles has a base of $ \frac{60}{2} = 30 $ m. Using the Pythagorean theorem:

["# Finding the Height of a Trapezoid with Given Dimensions", "Understanding the height of a trapezoid is essential in geometry, especially when calculating area or solving real-world problems involving land shape, construction, or design. In this article, we’ll explore how to determine the height of a trapezoid with known base lengths and leg lengths using the Pythagorean theorem.", "## The Trapezoid Configuration", "We consider a trapezoid with:\n- Shorter base length $ a = 120 $ m\n- Longer base length $ b = 180 $ m\n- Equal legs of length $ l = 50 $ m", "When perpendiculars are dropped from the endpoints of the shorter base $ a $ to the longer base $ b $, two right triangles form on the sides and a rectangle at the center. The difference in the base lengths is $ b - a = 60 $ m. Since the left and right sides are equal, each right triangle has a base of $ 30 $ m — half of 60 m.", "## Applying the Pythagorean Theorem", "Let $ h $ represent the height of the trapezoid — the length of the perpendicular dropped from the shorter base to the longer base.", "In each of the right triangles formed:\n- One leg is $ h $, the height\n- The other leg is $ 30 $ m\n- The hypotenuse is the leg of the trapezoid, $ l = 50 $ m", "By the Pythagorean theorem:\n$$\nh^2 + 30^2 = 50^2\n$$", "Now compute:\n$$\nh^2 + 900 = 2500\n$$\n$$\nh^2 = 2500 - 900 = 1600\n$$\n$$\nh = \sqrt{1600} = 40 , \ ext{meters}\n$$", "## Conclusion", "Thus, the height of the trapezoid with bases 120 m and 180 m and equal legs of 50 m is 40 meters. This geometric solution simplifies finding vertical measurements in irregular trapezoids and supports applications like land surveying and architectural design.", "For efficient problems involving right triangles within trapezoids, drop perpendiculars, isolate the right triangle, and apply:\n$$\nh = \sqrt{l^2 - \left(\frac{b - a}{2}\right)^2}\n$$\nThis formula ensures accurate height computation with minimal steps."]