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Parallelogram A B C D is shown. Line segment X Y goes from point X on side A B to point Y on side C D to form 2 trapezoids.
Figure ABCD is a parallelogram. Two trapezoids are created using line segment XY such that AX ≅ CY.
What is true about the areas of the trapezoids?
Each area is equal to half of the area of ABCD.
The area of AXYD is less than the area of BXYC.
The area of AXYD is greater than the area of BXYC.
Each area is equal to the area of ABCD.
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Each area is equal to half of the area of ABCD.
Reason: Put A=(0,0), B=(w,0), D=(u,h), C=(w+u,h). Let X=(t,0) on AB and, since AX = CY = t, let Y=(u+w−t,h) on CD. Using the shoelace formula for A–X–Y–D gives
Area(AXYD) = 1/2 |(u+w)h − u h| = 1/2 · w h,
and the parallelogram's area is w h, so Area(AXYD) = 1/2·Area(ABCD). The other trapezoid BXYC is the remainder, so it also has half the parallelogram's area.
Reason: Put A=(0,0), B=(w,0), D=(u,h), C=(w+u,h). Let X=(t,0) on AB and, since AX = CY = t, let Y=(u+w−t,h) on CD. Using the shoelace formula for A–X–Y–D gives
Area(AXYD) = 1/2 |(u+w)h − u h| = 1/2 · w h,
and the parallelogram's area is w h, so Area(AXYD) = 1/2·Area(ABCD). The other trapezoid BXYC is the remainder, so it also has half the parallelogram's area.
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