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

The Logical approach to this question starts with the understanding that the parallelogram is actually a rhombus, since all of its sides are equal. In order to find the area of a rhombus we need either its diagonals (like and kind of deltoid) or a side and a height.
Statement (1) gives us a 45 degrees angle. Since the height creates a right triangle in which the hypotenuse is one of the sides, this triangle is a 45-45-90 triangle, and thus one side is enough in order to find all others. So we have a side and a height - that's enough!
Statement (2) gives us this exact same height, so it is also sufficient on its own.
The correct answer is (D).

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Since we are given a parallelogram with all equal sides, it’s actually a rhombus and we need to determine the area of the rhombus.

Statement One Alone:

One angle of P measures 45 degrees.

This is enough to determine the area of P. For example, we can draw a diagonal that doesn’t intercept the 45-degree angle and create two congruent triangles, each with two sides of length 1 and the included angle of 45 degrees. Knowing two sides and an included angle is sufficient to determine the area of the triangle since that triangle is unique. Multiplying that area by 2 will give us the area of the parallelogram/rhombus. So statement one alone is sufficient to answer the question.

Statement Two Alone:

The altitude of P is √2/2.

This is enough to determine the area of P. Since one side of P can be used as the base of the parallelogram/rhombus, knowing the altitude/height of P allows us to find the area;

base x height = 1 x √2/2 = √2/2

Answer: D

Hi All,

We're told that each side of parallelogram P has length 1. We're asked for the area of P. This is a great 'concept question', meaning that if you know the concepts involved, then you don't actually have to do much math to get the correct answer. Here, since we know all 4 sides of the parallelogram, if we have ANY of the 4 angles, then can determine the area.

(1) One angle of P measures 45 degrees.

A parallelogram is 360 degrees and 'opposite' angles are equal. With the information in Fact 1, we know that the 4 angles are 45/135/45/135. Combined with the side lengths (which we know are all 1s), we can determine the exact area of this shape.
Fact 1 is SUFFICIENT

(2) The altitude of P is √2/2

The information in Fact 2 requires a bit more work, but we can now draw 2 RIGHT triangles "inside" the parallelogram. Each of those right triangles would have a hypotenuse of 1 (since the side lengths are 1s) and a 'height' of √2/2. We could determine the exact value of the 3rd side and the two angles (it ends up being a 45/45/90 right triangle), so we can then determine the area of the overall shape.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Statement 1
Area of P= 1*1*sin(45)= 1/(2)^{1/2}

Statement 2
Area of P= Altitude*Base= [(2)^{1/2}]/2 *1=1/(2)^{1/2}

IMO D

Is the square not a parallelogram as well? How do we arrive at the conclusion that this shape is a rhombus and not a square?

Square is also a rhombus (special case).

You do not have to figure out the name of the particular quadrilateral. As long as you know the formula for area of a parallelogram, you can answer the question.

This is a DS question. There is no need to actually calculate areas and other variables.

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Thank you gmatpd. The formulas for area differ for different types of paralleograms?

I just realized st. 1 mentions the angle=45, definitely cannot be a square.

How do you know that it's a 45-45-90 triangle?

Hi, would you mind clarifying how you get to V2/2, please?

So, I understand it is an isosceles triangle, and that height=base.
Also, you could use either the 45-45-90 triangle proportion where 1= xV2
Or Pythagorean where 1^2= h^2 + h^2.... can you explain how do you proceed from there to obtain V2/2?

Thank you

Hi

Furthering what cartiro has asked. Please may you explain how do you get from 1/√2 to 2√2 ?

I understand that 1/√2 * 1 = 2√2. My question is how do you get to that number without using a calculator?

45:45:90 triangle =

X:X:X[square_root]2[2/square_root]

1 = X[square_root]2[2/square_root]

X = 1/[square_root]2[2/square_root]

How come the short leg of X become a [square_root]2[2/square_root]/2 ?

There are two types of parallelogram that have four equal sides: a rhombus and a square. There are thus two possible solutions.

S1: Therefore, it must the rhombus. We can use trigonometry to find the height and then find the area. SUFFICIENT.

S2: This measurement can only be correct if it's referring to the height of the rhombus. SUFFICIENT.

The AREA OF RHOMBUS = BASE * HEIGHT (Not 1/2*BASE*HEIGHT - That's the formula for area of triangle).

The parallelogram is a rombus so
Ans is D.

(1) We get 45-45-90 right triangle, which gives altitude \(\frac{\sqrt{2}}{2}\).; Sufficient.

(2) Altitude is already given of the rhombus. Sufficient,.

The answer is D.

Answer: Option D

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