Proving That ABCD Is a Parallelogram

In geometry, a parallelogram is a quadrilateral with opposite sides that are parallel to each other. To prove that a given shape is a parallelogram, we need to demonstrate this property. In this article, we will explore one such example and see how we can use various geometric concepts to establish the fact.

The problem at hand involves a rectangle DEFG, where DC = 8√5. We are also given two statements: Statement #1 claims that the area of the rectangle is 8√5, while Statement #2 states that the altitude AH of parallelogram ABCD is 5. Our task is to determine which statement alone is sufficient to prove that ABCD is a parallelogram.

Let's start by analyzing Statement #1: The area of rectangle DEFG is indeed 8√5. Since we know that the area of a parallelogram is equal to the base multiplied by its altitude, this means that DC = EQ (since EQ * DC = 8√5). This alone does not necessarily mean that ABCD is a parallelogram, as there could be other shapes with the same property.

Now let's examine Statement #2: Line AH, the altitude of parallelogram ABCD, is 5. Unfortunately, this statement alone is insufficient to prove that ABCD is a parallelogram. We would need additional information about the base or height of the parallelogram to establish its properties.

However, when we combine both statements, the picture becomes clearer. Using Statement #1, we know that EQ * DC = 8√5, and from Statement #2, we have AH = 5. Now, consider a rectangle DEFG with CP perpendicular to DE, where P is on DE. We can draw a diagonal line from C to E, dividing the rectangle into two triangles: DEC and CFG.

Using the fact that DC = EQ, we can rewrite EQ * DC as CP * DE. Since DEFG is a rectangle, CP = EF (as CP is perpendicular to DE). This means that the area of triangle DEC is equal to 1/2 * CP * DE. Now, using Statement #1 again, we have EQ * DC = 8√5, which implies that Area DEC = 1/2 * CP * DE = 4√5.

Since we know that the area of a parallelogram is equal to its base multiplied by its altitude, we can now conclude that ABCD is indeed a parallelogram. This is because EQ * DC = CP * DE, which means that the area of triangle DEC (and therefore the entire parallelogram) is 4√5.

In conclusion, while Statement #1 alone does not prove that ABCD is a parallelogram, combining it with Statement #2 allows us to establish this fact. This problem serves as an excellent example of how geometric concepts can be used to solve complex problems on the GMAT, and highlights the importance of paying close attention to detail when working through difficult geometry questions.

Prove that abcd is a parallelogram