geometry Q

[BodybldgDoc]

How do we solve this?

If the diagonal BD of a square ABCD is 3. What is the perimeter of the square?
a. 4.5
b. 12
c. 3 sq rt 2
d. 6 sq rt 2
e. 12 sq rt 2

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[Lonely Sol]

OKay you know its a square so the sides are all the same

Since the diagonal forms 2 right triangles, so you know you can use the Pythagorean Theorem: a^2+ b^2=c^2.
since a=b because all sides of the square are equal
2a^2=3^2=9
a^2=9/2
a=3/sqrt(2)

So the perimeter is 4*(3/sqrt(2)) = 12/sqrt (2) =6 sqrt(2)



Another way is to remember the 45-45-90 ratio, since if you cut the square diagonally, it will form 2 - 45-45-90 triangles.

45-45-90, then sides is x , x , x sqrt(2)

since the hypotenouse is 3, then the sides will be 3/sqrt(2)
4 sides, so 4*(3/sqrt(2)) = 12/sqrt (2) =6 sqrt(2)

 

[Speeddater]

How do we solve this?

If the diagonal BD of a square ABCD is 3. What is the perimeter of the square?
a. 4.5
b. 12
c. 3 sq rt 2
d. 6 sq rt 2
e. 12 sq rt 2

Hey Bodybldgdoc, how's the studying coming along? I'm on NYU & Columbia's waitlist and I dread the Qr & PAT, I'm praying endlessly that I get in & don't have to re-take. Anyway, I arrived at ans d)

Drawing a diagram is very helpful. Here's the recipe for arriving at the answer:

1) Draw & label square ABCD

2) For squares, all four sides are equal, you'll see how this helps us in a minute.

3) The diagonal divides the square into two right triangles / isoceles triangles. This is true because all four sides of a square are equal. Note the perimeter is the sum of all four sides: P = 4s and if you remember your 45-45-90 right triangle triplets, the two legs adjacent to the hypotenuse are equal in angle and in length. The hypotenuse is the side of those legs multiplied by sq rt 2.

Let the side of the square = s
So BD = s sq rt. 2

S = BD / sq. rt 2 = 3 / sq. rt. 2, and now rationalize the denominator by multiplying the fraction by (3 / sq. rt. 2) x (sq. rt. 2) = s = 3 sq. rt 2 / 2

4) Remember our perimeter formula we established earlier? Well now we need to consult it: P = 4s & its simply plug & chug from here:



P = 4s = 4(3 sq. rt. 2 / 2) = 6 sq. rt. 2 - choice d

Hope this helps, & happy studying