Question

as shown in the figure below, the side length of square abcd is 8. on each side, segments ae = bf = cg = dh = 6 are cut. the area of quadrilateral efgh is __. a. 34 b. 36 c. 40 d. 100 in the figure, in trapezoid abcd, $overline{ad}paralleloverline{bc}$, $overline{bd}$ is the diagonal, and the median $overline{ef}$ intersects $overline{bd}$ at point o. if fo - eo = 3, then bc - ad is equal to __. a. 4 b. 6 c. 8 d. 10

Explanation:

Step1: Calculate area of square ABCD

The area of square \(ABCD\) with side - length \(a = 8\) is \(S_{ABCD}=a^{2}=8^{2}=64\).

Step2: Calculate area of four right - angled triangles

In right - angled triangle \(AEH\), \(AE = 6\), \(AH=8 - 6 = 2\).
The area of right - angled triangle \(S_{\triangle AEH}=\frac{1}{2}\times AE\times AH=\frac{1}{2}\times6\times2 = 6\).
Since \(\triangle AEH\cong\triangle BFE\cong\triangle CGF\cong\triangle DHG\) (by Side - Angle - Side congruence criterion as the angles of the square are right - angles and the corresponding side - lengths are equal), the total area of the four right - angled triangles is \(S_{total\ triangles}=4\times6 = 24\).

Step3: Calculate area of quadrilateral EFGH

The area of quadrilateral \(EFGH\) is \(S_{EFGH}=S_{ABCD}-S_{total\ triangles}\).
\(S_{EFGH}=64 - 24=40\).

for second question:

Step1: Recall the property of the median of a trapezoid

The median \(EF\) of trapezoid \(ABCD\) with \(AD\parallel BC\) has the property that \(EF=\frac{AD + BC}{2}\), and also, since \(AD\parallel EF\parallel BC\), in \(\triangle ABD\), \(E\) is the mid - point of \(AB\) and \(EO\parallel AD\), then \(EO=\frac{1}{2}AD\) (by the mid - point theorem in a triangle). Similarly, in \(\triangle BCD\), \(FO=\frac{1}{2}BC\).

Step2: Use the given condition \(FO - EO=3\)

Substitute \(FO=\frac{1}{2}BC\) and \(EO=\frac{1}{2}AD\) into \(FO - EO = 3\).
We get \(\frac{1}{2}BC-\frac{1}{2}AD = 3\).
Factor out \(\frac{1}{2}\): \(\frac{1}{2}(BC - AD)=3\).

Step3: Solve for \(BC - AD\)

Multiply both sides of the equation \(\frac{1}{2}(BC - AD)=3\) by 2, we get \(BC - AD=6\).

Answer:

C. 40