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

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If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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26 Apr 2019, 02:40
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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 $$\frac{\sqrt{2}}{2}$$.

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Re: If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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26 Apr 2019, 05:09
If each side of parallelogram P has length 1, what is the area of P ?
We are looking at a RHOMBUS. The area would depend on an angle..
opposite angles are EQUAL and the sum of two angles is 180.
So just knowing one angle is sufficient

(1) One angle of P measures 45 degrees.
So, when we draw an altitude, it becomes an isosceles right angled triangle with hypotenuse as 1, so altitude=side ..
$$s^2+s^2=1^2....s^2=\frac{1}{2}$$ or side=altitude=$$\frac{\sqrt{2}}{2}$$
Area = 1*$$\frac{\sqrt{2}}{2}$$=$$\frac{\sqrt{2}}{2}$$.

(2) The altitude of P is $$\frac{\sqrt{2}}{2}$$.
Area = 1*$$\frac{\sqrt{2}}{2}$$=$$\frac{\sqrt{2}}{2}$$.

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Re: If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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27 Apr 2019, 08:51
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.

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Re: If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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07 May 2019, 18:35
Bunuel wrote:
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 $$\frac{\sqrt{2}}{2}$$.

DS28502.01
OG2020 NEW QUESTION

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

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Re: If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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11 May 2019, 14:18
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

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Re: If each side of parallelogram P has length 1, what is the area of P ?  [#permalink]

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11 May 2019, 16:52
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
Re: If each side of parallelogram P has length 1, what is the area of P ?   [#permalink] 11 May 2019, 16:52
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# If each side of parallelogram P has length 1, what is the area of P ?

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