High School

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In rectangle ABCD, points X and Y are the midpoints of AD and DC, respectively. Lines BX and CD, when extended, intersect at E. Lines BY and AD, when extended, intersect at F. If the area of ABCD is 60, then what is the area of triangle BEF?

A. 60
B. 80
C. 90
D. 190

Answer :

Final answer:

The area of triangle BEF inside rectangle ABCD, with X and Y being the midpoints of sides AD and DC, and lines BX and BY extending to meet CD and AD at E and F respectively, is 60 units. The Correct Answer is Option.A.

Explanation:

To find the area of triangle BEF in a rectangle ABCD, first we note that points X and Y are the midpoints of AD and DC, respectively.

Since BX and BY are diagonals and we extend them to meet CD and AD at points E and F respectively, triangles BXE and BYF each cover half of the rectangle's area.

The rectangle has an area of 60 units, meaning that each triangle within it has an area of 30 units.

Now, consider triangle BED and triangle BFC, which are also formed by the diagonal lines.

Since X and Y are midpoints, these triangles are congruent to triangles BXE and BYF respectively and therefore each have an area of 30 units as well.

The area of triangle BEF is equal to the sum of the areas of triangles BED and BFC, which is 60 units.

Conclusion:

The area of triangle BEF is 60 units, which matches option A of the multiple-choice question presented by the student.

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