Question

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

Explanation:

Step1: Recall properties of parallelogram

In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AB = CD\). Since \(\overline{AX}\cong\overline{CY}\), then \(BX=DY\) (because \(AB - AX=CD - CY\)).

Step2: Analyze trapezoids' heights

The height \(h\) of trapezoid \(AXYD\) and trapezoid \(BXYC\) is the same as the height between the parallel - sides \(AB\) and \(CD\) of parallelogram \(ABCD\).

Step3: Use trapezoid area formula

The area formula 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.
For trapezoid \(AXYD\), let the parallel sides be \(AX\) and \(DY\), and height \(h\). For trapezoid \(BXYC\), let the parallel sides be \(BX\) and \(CY\), and height \(h\).
Since \(AX + DY=BX + CY\) (because \(AX = CY\) and \(BX = DY\)) and they have the same height \(h\), the areas of the two trapezoids are equal.
Also, the sum of the areas of trapezoid \(AXYD\) and trapezoid \(BXYC\) is equal to the area of parallelogram \(ABCD\). So each area is equal to half of the area of \(ABCD\).

Answer:

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