Finding proportions of simple shapes using algebra

Simonsky

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The question I'm asking about is question 3.c here:

proportions.jpg

The question expects you to work it out using algebra even though it's easy to see that the triangles in the trapezium are congruent. From this I can see that 2/3 of the trapezium is shaded-but I can't seem to get this using algebra. here's what I tried:

Are of trapezium = 1/2(a+b)h

The area of the combined shaded triangles will be - 2[1/2(1/2bh)] =2(1/4bh) = 1/2bh. So the proportion will be:

(1/2bh)1/(2ah+1/2bh)

can't get this to equate to 2/3 ! Clearly something faulty I can't see (but will result in me slapping my forehead shouting out dooooh! when someone points it out.

Help appreciatively received.
 

HallsofIvy

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For 3c you have three right triangles, all with the same base and height so all with the same area. Two of them are shaded. Without doing any more calculations, isn't it obvious that 2/3 of the area is shaded?
 

Simonsky

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For 3c you have three right triangles, all with the same base and height so all with the same area. Two of them are shaded. Without doing any more calculations, isn't it obvious that 2/3 of the area is shaded?

I agree-that's what I said above but the question I asked is about representing that with the algebra.
 

Dr.Peterson

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The question I'm asking about is question 3.c here:

View attachment 9709

The question expects you to work it out using algebra even though it's easy to see that the triangles in the trapezium are congruent. From this I can see that 2/3 of the trapezium is shaded-but I can't seem to get this using algebra. here's what I tried:

Are of trapezium = 1/2(a+b)h

The area of the combined shaded triangles will be - 2[1/2(1/2bh)] =2(1/4bh) = 1/2bh. So the proportion will be:

(1/2bh)1/(2ah+1/2bh)

can't get this to equate to 2/3 ! Clearly something faulty I can't see (but will result in me slapping my forehead shouting out dooooh! when someone points it out.

Help appreciatively received.
Did you mean to write (1/2 bh)/[1/2 ah + 1/2 bh] ? What you wrote doesn't make sense.

But you also need to take into account that b = 2a.

I might start with that, so that each triangle is ah/2, making a total of ah, while the trapezoid is (a+2a)h/2.

By the way, it's best to avoid writing 1/2bh, which could be read as 1/(2bh) rather than, as you intend, (1/2)bh. I at least put in spaces, 1/2 bh, but I prefer writing it as a division, putting the 2 last, bh/2, for clarity.
 

HallsofIvy

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Personally, I would say that observing that the three triangles are identical so that the area of two of them is 2/3 of the entire area is perfectly good answer. If you want more detail then call the height of the trapezoid, which is also the height of each triangle, "h". The bases are all labeled as being equal- call that "b" so the area of each triangle is (1/2)bh and the entire area is 3(1/2)bh= (3/2)bh. The area of two of them is bh and the shaded area is \(\displaystyle \frac{bh}{(3/2)bh}= \frac{2}{3}\). Seems a bit silly to have to say all that.
 
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Dr.Peterson

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The question expects you to work it out using algebra even though it's easy to see that the triangles in the trapezium are congruent. From this I can see that 2/3 of the trapezium is shaded-but I can't seem to get this using algebra.
I have to agree with the others: the question doesn't mention algebra, the title at the top of the page is "reasoning and problem solving", not "algebra", and reasoning about congruent shapes is far simpler and more satisfying than algebra. On what grounds do you say "The question expects you to work it out using algebra"? Did your teacher say that? Does the book say that somewhere else?

It's fine if you just wanted to see what could be done with algebra, knowing already that it wasn't necessary -- that kind of "what if" is how people learn math well. Just be sure you fully understand that algebra is no more mathematical and valuable than other kinds of reasoning.
 

Simonsky

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Did you mean to write (1/2 bh)/[1/2 ah + 1/2 bh] ? What you wrote doesn't make sense.

But you also need to take into account that b = 2a.

I might start with that, so that each triangle is ah/2, making a total of ah, while the trapezoid is (a+2a)h/2.

By the way, it's best to avoid writing 1/2bh, which could be read as 1/(2bh) rather than, as you intend, (1/2)bh. I at least put in spaces, 1/2 bh, but I prefer writing it as a division, putting the 2 last, bh/2, for clarity.
Ooooo.K! I didn't notice that b = 2a- full marks to me for lack of observational skills! (Is maths about noticing the patently obvious sometimes?)

So, my cumbersome algebra becomes: 1/2 (2ah)/ 1.5ah = 1/3

Thanks for that help Dr. Peterson-the question for me to ask myself is: 'why don't I notice these obvious things?' Thinking mathematically will elude me unless I can improve in this area.
 

Denis

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Simon, go stand in the corner for 15 minutes...:rolleyes:
 

Dr.Peterson

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Ooooo.K! I didn't notice that b = 2a- full marks to me for lack of observational skills! (Is maths about noticing the patently obvious sometimes?)

So, my cumbersome algebra becomes: 1/2 (2ah)/ 1.5ah = 1/3

Thanks for that help Dr. Peterson-the question for me to ask myself is: 'why don't I notice these obvious things?' Thinking mathematically will elude me unless I can improve in this area.
One thing to do is to make the "obvious" obvious!

I would have started by labeling the three congruent segments as x or a or whatever, implicitly defining that variable and also reminding myself what lengths are known to be equal.

Another valuable skill is to take time after solving a problem to look through what you did as an outside observer, first to see if everything is right, and then to see if you could have done something differently to make it easier. Then a third time to allow what you learned from it to stick.
 
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