Source: Official Guide Revised GRE 1st Ed. Geometry Exercises; #11

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# In parallelogram ABCD below, find the following

In parallelogram ABCD below, find the following.
(a) Area of ABCD
(b) Perimeter of ABCD
(c) Length of diagonal BD

### 4 Explanations

5

Lindsey Capano

I am still very confused on how to solve part C...is there anyway someone can explain it?

Aug 26, 2018 • Comment

Sam Kinsman, Magoosh Tutor

Hi Lindsey,

Happy to help!

The first step is to draw a line from point B to line AD, such that the line makes a right angle with the line AD. We'll call the place where it meets AD point P. And the line we've drawn is BP.

Now, we know that the height of the parallelogram is 4. So we know that line BP has a length of 4.

Next, look at the triangle on the far right of the diagram. There's a point below point C in the diagram (where there's a right angle). Let's call that point X. We know that triangle DCX is a right triangle, with sides 2 and 4. We also know that line CD is parallel to line AB, and that line CD has the same length as AB. Thus, we know that triangle DCX is congruent with triangle ABP.

Therefore, we know that the length of AP is 2. And thus, PD has a length of 10 (because AP + PD = 12).

So now, we know that PD = 10, and BP = 4. Together, these lines make a right triangle: BPD. And the hypotenuse of this triangle is BD.

BD is the length we are being asked for. So all we have to do now is use the pythagorean theorem:

10^2 + 4^2 = BD^2
100 + 16 = BD^2
116 = BD^2

sqrt (116) = BD
sqrt (4 * 29) = BD
2 * sqrt (29) = BD

I hope that helps! :)

9

amirhossein koleini

I know the answer 2sqrt(29) is true but I have a question:
bd=ac
ACO is Right triangle
12+2=14
so
c^2=14^2+4^2
c=2sqrt(53)
therefore:
BD:2sqrt(53)

Oct 21, 2016 • Comment

Sam Kinsman, Magoosh Tutor

Hi Amirhossein,

That's a good question! Although it may look like it, BD and AC are not equal to each other. The diagonals of a parallelogram are not always equal in length.

If you draw a line from point B to line AD, you'll see that line BD travels a horizontal distance of 10 units. Line AC, on the other hand travels a horizontal distance of 12 units. That's why AC and BD are not the same.

amirhossein koleini

thank you very much <3

16

Jonathan , Magoosh Tutor

Hi Alicia,
2sqrt(5) is just sqrt(20) simplified. For the GRE, it's important to be able to convert square roots into their simplified forms if necessary.

If the number inside the sqrt -- in this case, that number is 20 -- contains a perfect square as a factor, then we know we can simplify the root further. Here we have a 20, which can be written as 4 * 5. Since 4 is a perfect square, we know we can simplify further. How do we do this? Well, we use the key property of roots:

sqrt(A * B) = sqrt(A) * sqrt(B) so that

sqrt(20) can be written as

sqrt(4) * sqrt(5)

Now, sqrt(4) = 2, so we can write this as:

2sqrt(5).

5 doesn't contain any perfect squares as a factor (except for 1 of course) so we cannot simplify this further.

Now, do we HAVE to always express in simplified form? Not necessarily. But answer choices may often be in simplified form, so it's important to know how to simplify like this.

Also, in some case, we may want to add or subtract square roots. We cannot immediately add, for example:

4sqrt(5) + sqrt(45) But if we make the sqrt's the same, then we can add or subtract them. So here, since sqrt(45) = 3sqrt(5) we have:

4sqrt(5) + 3sqrt(5) = 7sqrt(5)

4.47 is not an exact value...it's rounded. There's nothing wrong with approximating in some cases, but we want to avoid relying on a calculator, so keep in mind converting to decimals could use up valuable time.

I hope this helps :)

Jan 19, 2015 • Comment

9

Angie Robinson

A) The area is found using the formula (base)(height). Both are given... (12)(4) = 48

B) The perimeter is found by adding the lengths of all sides. To find the length of CD use Pythagorean Thm... CD= sqrt(4^2+2^2) = 2sqrt(5)
CD = AB
AD + DC + CB + BA = 12 + 2sqrt(5) + 12 + 2sqrt(5) = 24 + 4sqrt(5)

C) Draw height line from angle B to AD. Call this point P. BP = 4. Using properties of parallel lines, angle A is equal to the outside angle D. Triangle ABP = Triangle DCO (outside point). So AP = 2 => PD = 10. Use Pythagorean Thm to find BD. BD = sqrt(4^2 + 10^2) = sqrt(116) = 2sqrt(29)

May 16, 2014 • Comment

Alicia Scott

Why is the length of CD written in the form 2sqrt(5), instead of sqrt(20) or 4.47? How do you get the answer in this form?

Sanchit Bogra

How did you find out the length of side AP?

Cydney Seigerman, Magoosh Tutor

Hi Sanchit :)

Happy to help! BP is perpendicular to AD, so triangle ABP is a right triangle. That means that we can find AP (the hypotenuse) by using the Pythagorean theorem and the lengths of the other two sides of the right triangle ABP.

AB = CD = 2
BP = 4

AP^2 = AB^2 + BP^2
AP^2 = 2^2 + 4^2
AP^2 = 4 + 16
AP^2 = 20
AP = sqrt(20) = 2sqrt(5)

Hope this helps :)

Raghav Bhandari

Thanks alot Angie, this ETS question - (Length of diagonal of a parallelogram) is kind of tough , I thought it would require me to revise trignometry to solve it, but your solution is neat and inspires creative thinking.

How can I say triangle bap= triangle cdo?

Jonathan , Magoosh Tutor

Hi Yawvaar,
BAP and CDO are both right triangles with equal angles and equal sides. We know the angles are equal because they are formed by the intersection of the base with two parallel sides of the parallelogram. We know the opposite sides of the parallelogram are equal as well. We also know the heights (the vertical/perpendicular distance) between the top and bottom parallel bases of the parallelogram are equal. So the triangles must be congruent.
I hope that helps.