Two similar trapezoids have the side length of 16m and 30m. If the smaller trapezoid is 325m, what is the area of the larger trapezoid?

Question

ashleyisakitty · Answer

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e.mccormick · Answer

OK, they say they are similar.  You know what that means?

ashleyisakitty · Answer

Kinda... I have to do an equation like AC/AB=DF/DE? 

I know thats for triangles but we could revise it for trapezoids? lol. honestly i dont know i suppose.

e.mccormick · Answer

Yes, that is an OK way of saying it.  Technically it means the ratios of the sides are the same.  So if one side is half the size of a corresponding side, all sides are half their corresponding sides.

OK, so that is one piece of info you need.

Now, do you know how to find the area of a trapezoid?

ashleyisakitty · Answer

A=(b1+b2)*h/2

e.mccormick · Answer

Yes.  Now, they gave you some of all of that for the larger one.  So, the first thing to do is to find the rest, or enough of it that you have what you need.

ashleyisakitty · Answer

okay.

so 325=h(b1+16)/2...??

I dont know how to find out the height or 2nd base

e.mccormick · Answer

So, lets start with what you do know:
325=(b1+16)*h/2

using small letters for the small trapezoid.

?=(B1+30)*H/2

For the large.  But, you have another number set:

16/30 = b1/B1 = b2/B2 = h/H
The ratio that is the same.

e.mccormick · Answer

So you can use the ratio to solve for things and do replacments in one equation or the other.  The goal being to get everything in either values of the large or of the small.

e.mccormick · Answer

That would give you two equations in two unknowns.

ashleyisakitty · Answer

A bit confused, sorry. Doesnt that leave us still with no info other than the original bases? Am i supposed to interpret the other ratios?

e.mccormick · Answer

We are looking for the big stuff, so let me do some replacments in the small with valyes from the big.

325=(b1+16)*h/2

16/30 = b1/B1 = b2/B2 = h/H

OK.

16/30=h/H
(16/30)H=h
therefore:
325=(b1+16)*(16/30)H/2

16/30=b1/B1
(16/30)B1=b1

therefore:
325=((16/30)B1+16)*(16/30)H/2

Now I have:

325=((16/30)B1+16)*(16/30)H/2
A=(B1+30)*H/2

Lot less unknowns there!

e.mccormick · Answer

Do you see how I used the ratio to replace things in that equation?

ashleyisakitty · Answer

uh... kinda

e.mccormick · Answer

a/b=c/d
(a/b)*d=(c/d)*d  <-- the d cancels on the right
(a/b)d=c

e.mccormick · Answer

Hmmm....   there is an easier way.  This is going the long way around.  Almost forgot this!

The ratios of the area can be used to find the ratios of the volume!

e.mccormick · Answer

If the ratio of the distance is a/b, the ratio of the area is a^2/b^2 and the ratio of the volume is a^3/b^3. You have the ratio of the lines: 16/30 or 30/16, depending. Well, that means you square them to find the ratio of the areas.

ashleyisakitty · Answer

square 16/30? 
...

e.mccormick · Answer

Yes, or simplify it then square it.

e.mccormick · Answer

Now because you used 16/30 that is the small over the lage.  That means the ratio of the areas will also be small over large.

16^2/30^2 = 325/A

ashleyisakitty · Answer

Sorry but i am just not following you at all

e.mccormick · Answer

OK.  if b is the measure of any line on the small and B is the measure of any line on the big, then the ratio of the lines is b/B or B/b.  Both work.  One is the ratio of small to big and the other is ratio of big to small.

THe ratio of the areas would then be a/A or A/a.  However, the ratio of the ares is the rati of the lines squared!  So:

b^2/B^2 = a/A
and
B^2/b^2 = A/a

e.mccormick · Answer

Well, any line and the corresponding line.

ashleyisakitty · Answer

ok...but we dont have the 2nd bases ;-;

sorry but ive never been good at ratios.. obviously. you dont need to waste anymore of your time but thank you for trying.

e.mccormick · Answer

It is not a waste to me.  Like I said, I went the wrong direction at the start.  Was a tad distracted at that moment.  

You only need one pair of corresponding sides to find the ratio of the sides.

The ratio of the areas is related to the ratio of the sides.

e.mccormick · Answer

Can I show you with some squares.  It is easier.

e.mccormick · Answer

Then I can also show you with numbers and you can see what I mean.

ashleyisakitty · Answer

If you'd like to, id appreciate it, but i am going to be studying other questions as well so i will be opening this tab every few minutes.

e.mccormick · Answer

Lets take a 4*4 square and an 8*8 square.  Now, because they are squares all the ratios are easy.  I am going to use the small b for any side of the small one and a big B for any side of the large.

b=4
B=8

Now, because I know the sides, the areas are easy:

4*4=16
a=16
8*8=64
A=64

But let me show you what I meant:

The ratio of b/B is 4/8.  4/8 is 1/2.  This means that the ratio of the sides is 1/2.

The ratio of the areas, because I did b/B is a/A.  If I know the small, that means 16/A.

The ratio of the areas is ALSO the ratio of the sides squared, or 1^2/2^2.  That is 1/4.

Because these are both ratios of the area I can say:
16/A=1/4

Now I just need to solve for A.  Now, because I have what I want to solve for on the bottom, the first thing is to multiply both sides by A:
(16/A)*A=(1/4)*A
16=(1/4)A

OK, now I just need to move the 1/4 over, which means multiply by the reciprocal of 4.
16*4=(1/4)*4A
16*4=A

Well, 16*4=64!

I got the area I should have.

You are doing the same thing!  You numbers are:

b=16
B=30

a=325
A=?


And sorry about the early confusion.

ashleyisakitty · Answer

The answer is 1,143?

e.mccormick · Answer

Yah, it rounds to that.

e.mccormick · Answer

1142.578125