Question

given △abc with parallel lines dividing \\( \overline { a b } \\) 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
a. the ratio of the area of triangle dbe to the area of trapezoid degf is \\( \frac { 1 } { 3 } \\)
(type an integer or a simplified fraction)
b. the ratio of the area of triangle dbe to the area of trapezoid fgca is \\( \frac { 1 } { 5 } \\)
(type an integer or a simplified fraction)
c. the ratio of the area of trapezoid degf to the area of trapezoid fgca is

Explanation:

Step1: Use the property of similar triangles

Since the lines are parallel and \(AB\) is divided into three congruent segments. Let the height of \(\triangle DBE\) be \(h_1\), the height of \(\triangle BFG\) be \(h_2\) and the height of \(\triangle ABC\) be \(h_3\). If the ratio of the sides of similar triangles is \(k\), the ratio of their areas is \(k^{2}\). For \(\triangle DBE\) and \(\triangle BFG\), the ratio of their sides \(k_1=\frac{1}{2}\), so \(A_{DBE}:A_{BFG}=1:4\). For \(\triangle DBE\) and \(\triangle ABC\), the ratio of their sides \(k_2 = \frac{1}{3}\), so \(A_{DBE}:A_{ABC}=1:9\).
Let \(A_{DBE}=x\). Then \(A_{BFG} = 4x\) and \(A_{ABC}=9x\).

Step2: Calculate the area of trapezoids

The area of trapezoid \(DEGF=A_{BFG}-A_{DBE}=4x - x=3x\). The area of trapezoid \(FGCA=A_{ABC}-A_{BFG}=9x - 4x = 5x\).

Answer:

The ratio of the area of trapezoid \(DEGF\) to the area of trapezoid \(FGCA\) is \(\frac{3x}{5x}=\frac{3}{5}\)