Answer: 512 cm^2
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Step-by-step explanation:
Often with area problems like this, the trick is to somehow draw in an extra line (or perhaps several) so you can break the irregular shape into something more familiar. In this case, only one extra line segment is needed.
This is a tricky problem if you don't know where to add on the extra line segment. I managed to find it through trial and error. This line connects the top left and bottom right corners of the quadrilateral. Refer to the drawing below. The red line splits the brown quadrilateral into 2 triangles.
The lower triangle has base 24 cm and height 20 cm. The area of this lower triangle is b*h/2 = 24*20/2 = 240 square cm.
The upper triangle has base 16 cm and height 34 cm. It might help to rotate the diagram so that the "16 cm" is completely horizontal. Note that the base and height are perpendicular to one another. The area of this upper triangle is b*h/2 = 16*34/2 = 272
From here, we add the two triangular areas: 240+272 = 512 cm^2