Question

given △abc with parallel lines dividing (overline{ab}) into three congruent segments as shown, compare the following areas.
a. triangle dbe and trapezoid degf
b. triangle dbe and trapezoid fgca
c. trapezoids degf and fgca
d. trapezoid degf and triangle abc
e. trapezoid fgca and triangle abc
f. triangle abc and trapezoid deca
(type an integer or a simplified fraction.)
d. the ratio of the area of trapezoid degf to the area of triangle abc is (\frac{1}{3})
(type an integer or a simplified fraction.)
e. the ratio of the area of trapezoid fgca to the area of triangle abc is (\frac{5}{9})
(type an integer or a simplified fraction )
f. the ratio of the area of triangle abc to the area of trapezoid deca is (square)
(type an integer or a simplified fraction.)

Explanation:

Step1: Use the property of similar triangles

Since the parallel lines divide \(AB\) into three congruent segments, let \(BD = DF=FA=a\). Let the height of \(\triangle BDE\) be \(h_1\), the height of \(\triangle BFG\) be \(h_2\), and the height of \(\triangle ABC\) be \(h\). By the property of similar triangles (if two triangles are similar, the ratio of their sides is equal to the ratio of their heights), for \(\triangle BDE\) and \(\triangle ABC\), \(\frac{BD}{BA}=\frac{h_1}{h}\). Since \(BD = a\) and \(BA=3a\), \(\frac{BD}{BA}=\frac{1}{3}\), so \(h_1=\frac{1}{3}h\). For \(\triangle BFG\) and \(\triangle ABC\), \(\frac{BF}{BA}=\frac{2}{3}\), so the height of \(\triangle BFG\) is \(h_2 = \frac{2}{3}h\).

Step2: Calculate the area of \(\triangle ABC\)

The area of \(\triangle ABC\), \(A_{ABC}=\frac{1}{2}\times BA\times h=\frac{1}{2}\times3a\times h=\frac{3}{2}ah\)

Step3: Calculate the area of trapezoid \(DECA\)

The area of \(\triangle BDE\), \(A_{BDE}=\frac{1}{2}\times BD\times h_1=\frac{1}{2}\times a\times\frac{1}{3}h=\frac{1}{6}ah\)
The area of trapezoid \(DECA=A_{ABC}-A_{BDE}=\frac{3}{2}ah-\frac{1}{6}ah=\frac{9ah - ah}{6}=\frac{4}{3}ah\)

Step4: Find the ratio

The ratio of the area of \(\triangle ABC\) to the area of trapezoid \(DECA\) is \(\frac{A_{ABC}}{A_{DECA}}=\frac{\frac{3}{2}ah}{\frac{4}{3}ah}\)

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Answer:

\(\frac{9}{8}\)