Question

figure abcd is a parallelogram. two trapezoids are created using line segment xy such that \\(\overline{ax} \cong \overline{cy}\\). what is true about the areas of the trapezoids? \\(\bigcirc\\) each area is equal to half of the area of abcd. \\(\bigcirc\\) the area of axyd is less than the area of bxyc. \\(\bigcirc\\) the area of axyd is greater than the area of bxyc. \\(\bigcirc\\) each area is equal to the area of abcd.

Explanation:

Step1: Recall parallelogram and trapezoid area

In a parallelogram \(ABCD\), \(AB \parallel CD\) and \(AB = CD\). The area of a parallelogram is \(A = base\times height\). For trapezoid \(AXYD\) and \(BXYC\), the height of both trapezoids is the same as the height of the parallelogram (since they are between the same parallel lines \(AB\) and \(CD\)).

Step2: Analyze the bases of trapezoids

Given \(\overline{AX} \cong \overline{CY}\). Let \(AX = CY = x\), \(AB = CD = y\). Then, for trapezoid \(AXYD\), the two bases are \(AX = x\) and \(AD' = CD - CY=y - x\)? Wait, no. Wait, \(AB\) and \(CD\) are equal. So for trapezoid \(AXYD\), the bases are \(AX\) and \(DY\), and for trapezoid \(BXYC\), the bases are \(BX\) and \(CY\). Since \(AB=AX + XB\) and \(CD = DY+ YC\), and \(AB = CD\), \(AX = CY\), so \(XB=DY\).

Step3: Calculate area of trapezoids

The area of a trapezoid is \(A=\frac{1}{2}(b_1 + b_2)h\), where \(b_1,b_2\) are the two parallel sides and \(h\) is the height.

For trapezoid \(AXYD\): \(b_1 = AX\), \(b_2=DY\), height \(h\) (same as parallelogram).

For trapezoid \(BXYC\): \(b_1 = BX\), \(b_2=CY\), height \(h\).

Since \(AX = CY\) and \(DY = BX\) (from \(AB = CD\) and \(AX = CY\)), then \(\frac{1}{2}(AX + DY)h=\frac{1}{2}(CY + BX)h\). Also, the area of parallelogram \(ABCD\) is \(AB\times h=(AX + XB)\times h\). And the sum of the areas of the two trapezoids is \(\frac{1}{2}(AX + DY)h+\frac{1}{2}(BX + CY)h=\frac{1}{2}(AX + BX + DY + CY)h=\frac{1}{2}(AB + CD)h\). But since \(AB = CD\), this is \(\frac{1}{2}(2AB)h=AB\times h\), which is the area of the parallelogram. So each trapezoid's area is half of the parallelogram's area.

Let's check the options:

• Option 1: Each area is equal to half of the area of \(ABCD\). This matches our calculation.
• Option 2: The area of \(AXYD\) is less than \(BXYC\). No, since their areas are equal.
• Option 3: The area of \(AXYD\) is greater than \(BXYC\). No, same reason.
• Option 4: Each area is equal to \(ABCD\). No, their sum is equal to \(ABCD\), so each is half.

Answer:

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