Question

figure abcd is a parallelogram. two trapezoids are created using line segment xy such that (overline{ax}congoverline{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

Explanation:

Step1: Recall the formula for the area of a trapezoid

The area of a trapezoid is \(A=\frac{(a + b)h}{2}\), where \(a\) and \(b\) are the lengths of the parallel sides and \(h\) is the height.

Step2: Analyze the trapezoids \(AXYD\) and \(BXYC\) in parallelogram \(ABCD\)

In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AB = CD\), \(AD\parallel BC\). Let the height of the parallelogram (distance between \(AB\) and \(CD\)) be \(h\).
For trapezoid \(AXYD\), the parallel sides are \(AX\) and \(DY\), and for trapezoid \(BXYC\), the parallel sides are \(BX\) and \(CY\). Since \(AX\cong CY\) and \(AB = CD\) (so \(BX=DY\)), and the height \(h\) is the same for both trapezoids (because they are between the same parallel lines \(AB\) and \(CD\)).
Let \(AX = CY=x\) and \(BX=DY = y\).
The area of trapezoid \(A_{AXYD}=\frac{(AX + DY)h}{2}=\frac{(x + y)h}{2}\)
The area of trapezoid \(A_{BXYC}=\frac{(BX + CY)h}{2}=\frac{(y + x)h}{2}\)
Also, the area of parallelogram \(A_{ABCD}=AB\times h=(x + y)h\)

Answer:

Each area is equal to half of the area of \(ABCD\)