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If each side of parallelogram P has length 1, what is the area of P ?

1. One angle of P measures 45 degrees.

2. The altitude of P is image.png.


    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

    C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

    D. EACH statement ALONE is sufficient.

    E. Statements (1) and (2) TOGETHER are NOT sufficient.


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答案:
D


Geometry Quadrilaterals; Area


The figure below shows parallelogram P, its altitude h, and an interior angle with degree measure θ. The area of a parallelogram is its base times its altitude. Therefore, the area of parallelogram P is (h)(1) = h. Determine the value of h.

image.png

1. Given that θ = 45, it follows that the right triangle in the figure with hypotenuse 1 and side length h is a 45°−45°−90° triangle. Therefore, the length of each of its legs isimage.png

 

times the length of its hypotenuse, and hence image.png; SUFFICIENT.

 

 

2. Given thatimage.png , it follows from the comments above that the area of P isimage.png ; SUFFICIENT.


The correct answer is D;each statement alone is sufficient.


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