In convex quadrilateral ABCD, diagonals AC and BD intersect at point E. The area of \trian… — GRE General Test

In convex quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . The area of $\triangle ABE$ is $4$ , the area of $\triangle BCE$ is $8$ , and the area of $\triangle CDE$ is $16$ . Which of the following statements must be true? Select all that apply.

The area of $\triangle DAE$ is $8$ .Answer
Quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ .Answer
The length of segment $CD$ is twice the length of segment $AB$ .Answer
D
The total area of quadrilateral $ABCD$ is $32$ .
E
The midpoint of diagonal $AC$ lies on segment $DE$ .

Answer

The correct statements are that the area of triangle DAE is 8, quadrilateral ABCD is a trapezoid with AB parallel to CD, and the length of segment CD is twice the length of segment AB.

The statements confirming that the area of triangle DAE is 8, that ABCD is a trapezoid with AB parallel to CD, and that CD is twice length AB are all derived using area ratio properties of intersecting diagonals and triangle similarity.

Step-by-Step Solution

Determine the area of triangle DAE using diagonal segment ratios.

Area of triangle DAE = 8.

Triangles sharing a base line have areas proportional to the segments into which the intersecting line divides that base line: AE/EC = Area(ABE)/Area(BCE) = 4/8 = 1/2, so Area(DAE) = (1/2) * 16 = 8.

Check parallelism of opposite sides AB and CD.

AB is parallel to CD, making ABCD a trapezoid.

Area(ABC) = 4 + 8 = 12 and Area(ABD) = 4 + 8 = 12. Triangles with equal areas on the common base AB must have equal heights, implying line CD is parallel to line AB.

Calculate side ratio CD / AB using similar triangles.

CD = 2 * AB.

Since AB || CD, triangle ABE is similar to triangle CDE. The ratio of their areas is 16/4 = 4, so the side length ratio CD/AB = sqrt(4) = 2.

Compute total area of quadrilateral ABCD and evaluate diagonal midpoint position.

Total area is 36 (not 32), and the midpoint of AC is at (1/2)AC from A, which is distinct from E at (1/3)AC.

Total area = 4 + 8 + 16 + 8 = 36. Since E divides AC in a 1:2 ratio, E is not the midpoint of AC.

Key Concept

Properties of convex quadrilaterals, area decomposition via diagonal ratios, trapezoid parallelism criteria, and triangle similarity.

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