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Quadrilateral ABCD is a parallelogram. Let E be a point on line AB, and let F be the intersection of lines DE and BC. The area of triangle EBF is 4, and the area of triangle EAD is 9. Find the area of parallelogram ABCD.

 Apr 19, 2024
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Here's another way to solve this problem, focusing on proportionality within the parallelogram:

 

Proportions in Similar Triangles:

 

Since ABCD is a parallelogram, lines AD and BC are parallel. When line DE intersects line BC at point F, it creates two transversal lines. Because of this, corresponding angles on alternate sides of DE are congruent (alternate interior angles)

 

Therefore, triangles EAD and EBF are similar by Angle-Angle Similarity (AA).

 

Since the triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths:

 

Area(EBF) / Area(EAD) = (length(BF) / length(AD))^2

 

Given Information:

 

We are given that the area of triangle EBF is 4 and the area of triangle EAD is 9. Plugging these values into the equation:

 

4 / 9 = (length(BF) / length(AD))^2

 

Proportionality in a Parallelogram:

 

In a parallelogram, opposite sides have the same length. Therefore, length(BF) = length(AE) and length(AD) = length(BC). Substituting these equalities into the equation from step 2:

 

4 / 9 = (length(AE) / length(BC))^2

 

Area of Parallelogram:

 

The area of a parallelogram is equal to the base times the height. Since AD is parallel to BC, the height of parallelogram ABCD is the same as the height of triangle EAD (considering side AE as the base). We can express this using proportionality:

 

Area(ABCD) ∝ length(BC) * height(EAD)

 

From step 3, we know the ratio between the base of triangle EAD (which is also a base of the parallelogram) and side BC of the parallelogram:

length(AE) / length(BC) = √(4/9) (taking the square root of both sides)

 

Combining Proportions:

 

Since the area of the parallelogram is proportional to the product of its base and height, and we know the ratio between the base and a side of the parallelogram, we can combine these proportions:

 

Area(ABCD) ∝ (√(4/9)) * height(EAD)

 

Note: We don't need to find the actual height of triangle EAD since it cancels out when solving for the relative area of the parallelogram.

 

Relative Area of Parallelogram:

 

Since the area of triangle EAD is a constant value (9) and the other factor in the proportion is a constant resulting from the given information, the area of parallelogram ABCD is also proportional to a constant value.

 

Therefore, relative to the area of triangle EAD, the area of parallelogram ABCD is:

 

Area(ABCD) ∝ √(4/9) * Area(EAD) = √(4/9) * 9 = 4

 

Scaling to Actual Area:

 

While the previous step gives us the relative area compared to triangle EAD, we can find the actual area by considering that the area of triangle EBF is given as 4.

 

Since triangles EAD and EBF are similar, the ratio of their areas is the square of the ratio of their corresponding side lengths (as established in step 1). Therefore, the area of triangle EAD is 4 times the area of triangle EBF:

 

Area(EAD) = 4 * Area(EBF) = 4 * 4 = 16

 

Now, we can use the relative area we found in step 6 to solve for the actual area of the parallelogram:

 

Area(ABCD) = √(4/9) * Area(EAD) = √(4/9) * 16

 

Area(ABCD) = 4 * 4 = 16​.

 Apr 20, 2024

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