Find the Height First

Part of the Polygons & Quadrilaterals section of Coddy's Geometry journey. Lesson 42 of 59.

Often the height is not given, but a slanted leg is. Drop the height from the end of the leg: it makes a right triangle whose hypotenuse is the leg. The Pythagorean theorem then gives the height.

$Trapezoid with leg 13, the foot of the height 5 from A, and height h$

Here the leg AD is 13 and E is 5 from A, so ${h}^{2}+{5}^{2} = {13}^{2}$ , which gives ${h}^{2} = 144$ and h = 12. Only then does the area rule have all it needs: ½ × (16 + 6) × 12 = 132. Two rules in a row: Pythagorean theorem, then Area of a trapezoid.

Challenge

ABCD is a trapezoid with AB ∥ DC, AB = 11, DC = 5 and the leg AD = 5. The dashed height DE meets AB at E, 3 from A. Find the area of ABCD, then press Check.

Try it yourself

$Find the marked part of the figure$