# How to solve this Shaded Area Problem?

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Hi all, can someone show the detailed solution for this question? Rather challenging. Thanks in advance!

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#2

(Original post by

Hi all, can someone show the detailed solution for this question? Rather challenging. Thanks in advance!

**Darren888**)Hi all, can someone show the detailed solution for this question? Rather challenging. Thanks in advance!

What can you determine / what is difficult for you?

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#4

(Original post by

Sorry solutions are not allowed.

What can you determine / what is difficult for you?

**mqb2766**)Sorry solutions are not allowed.

What can you determine / what is difficult for you?

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(Original post by

Am I missing something? There doesn't seem any way to determine the actual position of the bottom right hand corner of the unshaded triangle, which would seem to make the problem impossible.

**DFranklin**)Am I missing something? There doesn't seem any way to determine the actual position of the bottom right hand corner of the unshaded triangle, which would seem to make the problem impossible.

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#6

It is useful to notice that the triangle made from AB and the vertex closest to E is similar to the one created by the intersection of AE with DC the point C and also the vertex closest to E. Also the fact that the provided side lengths are said to be of squares.

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#7

(Original post by

Thanks, that thought crossed my mind too.

**Darren888**)Thanks, that thought crossed my mind too.

Or if for some reason, it's independent of the position, pick a value that makes the calc easy.

Last edited by mqb2766; 4 weeks ago

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#8

(Original post by

Can you not set up the position as a variable, then equate areas to find it?

Or if for some reason, it's independent of the position, pick a value that makes the calc easy.

**mqb2766**)Can you not set up the position as a variable, then equate areas to find it?

Or if for some reason, it's independent of the position, pick a value that makes the calc easy.

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#9

(Original post by

What areas are you saying need to be equal?

**DFranklin**)What areas are you saying need to be equal?

Big rectangle = 3 shaded areas + unshaded triangle.

All could be worked out in terms of x (or independent of it).

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**mqb2766**)

Can you not set up the position as a variable, then equate areas to find it?

Or if for some reason, it's independent of the position, pick a value that makes the calc easy.

I just printed out a fresh copy. Will try again. 🙂

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#11

(Original post by

If x was the position of the bottom unshaded triangle point

Big rectangle = 3 shaded areas + unshaded triangle.

All could be worked out in terms of x (or independent of it).

**mqb2766**)If x was the position of the bottom unshaded triangle point

Big rectangle = 3 shaded areas + unshaded triangle.

All could be worked out in terms of x (or independent of it).

[I did momentarily wonder if it was one of those "everything cancels" problems where changing x doesn't actually change the area, but that turns out not to be the case. I think they just messed up the diagram and the bottom right corner of the triangle is supposed to be at E to be honest].

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#12

(Original post by

I think introducing a variable for an unmarked position, and giving an answer in terms of that variable is a bit of a stretch. particularly given the way the question is worded. Note that it not only shows you the lengths of ABCD and EFGH in the diagram, but expressly tells you in words as well when this isn't needed. It would seem very strange to jump from that "hand-holding" to "oh, but you need to decide for yourself to put in a variable for the position of the bottom RH corner and give an answer in terms of that variable".

[I did momentarily wonder if it was one of those "everything cancels" problems where changing x doesn't actually change the area, but that turns out not to be the case. I think they just messed up the diagram and the bottom right corner of the triangle is supposed to be at E to be honest].

**DFranklin**)I think introducing a variable for an unmarked position, and giving an answer in terms of that variable is a bit of a stretch. particularly given the way the question is worded. Note that it not only shows you the lengths of ABCD and EFGH in the diagram, but expressly tells you in words as well when this isn't needed. It would seem very strange to jump from that "hand-holding" to "oh, but you need to decide for yourself to put in a variable for the position of the bottom RH corner and give an answer in terms of that variable".

[I did momentarily wonder if it was one of those "everything cancels" problems where changing x doesn't actually change the area, but that turns out not to be the case. I think they just messed up the diagram and the bottom right corner of the triangle is supposed to be at E to be honest].

So either

* Spot the magic bit of geometry

* Slog it out with areas

* Decide the problem is wrong

Edit - going down the variable position - area approach seems to give a quadratic in x. So not too bad.

Last edited by mqb2766; 4 weeks ago

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#13

(Original post by

I tried to split the RHS triangle GFE, but to no avail in terms of coming up with the unknowns. My paper is a mess lol.

I just printed out a fresh copy. Will try again. 🙂

**Darren888**)I tried to split the RHS triangle GFE, but to no avail in terms of coming up with the unknowns. My paper is a mess lol.

I just printed out a fresh copy. Will try again. 🙂

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#14

**DFranklin**)

Am I missing something? There doesn't seem any way to determine the actual position of the bottom right hand corner of the unshaded triangle, which would seem to make the problem impossible.

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#15

(Original post by

Edit - going down the variable position - area approach seems to give a quadratic in x. So not too bad.

**mqb2766**)Edit - going down the variable position - area approach seems to give a quadratic in x. So not too bad.

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#16

(Original post by

It is useful to notice that the triangle made from AB and the vertex closest to E is similar to the one created by the intersection of AE with DC the point C and also the vertex closest to E. Also the fact that the provided side lengths are said to be of squares.

**Fufuabc123**)It is useful to notice that the triangle made from AB and the vertex closest to E is similar to the one created by the intersection of AE with DC the point C and also the vertex closest to E. Also the fact that the provided side lengths are said to be of squares.

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#17

(Original post by

I'd agree, but the question was raised about whether it's solvable and that's one way to do it and I agree about not being able to choose a trivial position, but it's one thing for the OP to check.

So either

* Spot the magic bit of geometry

* Slog it out with areas

* Decide the problem is wrong

Edit - going down the variable position - area approach seems to give a quadratic in x. So not too bad.

**mqb2766**)I'd agree, but the question was raised about whether it's solvable and that's one way to do it and I agree about not being able to choose a trivial position, but it's one thing for the OP to check.

So either

* Spot the magic bit of geometry

* Slog it out with areas

* Decide the problem is wrong

Edit - going down the variable position - area approach seems to give a quadratic in x. So not too bad.

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#18

(Original post by

I tend to agree with everything DFranklin said, but just for interest, a vector product approach yields a linear function in x.

**old_engineer**)I tend to agree with everything DFranklin said, but just for interest, a vector product approach yields a linear function in x.

But I think the most straightforward approach is to calculate the 3 shaded areas - finding the area of a triangle and trapezium (where you know the base and heights) is pretty easy.

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#19

It would help to know some context of the question also.

Is it in a specific section that could be used to hint as to the working out process? -that would help identify whether there is information missing (ie, question is wrong)

or if, in context there are assumptions that can be made that we wouldn’t make out of context

or whether variables should be introduced as suggested.

Is it in a specific section that could be used to hint as to the working out process? -that would help identify whether there is information missing (ie, question is wrong)

or if, in context there are assumptions that can be made that we wouldn’t make out of context

or whether variables should be introduced as suggested.

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#20

(Original post by

It would help to know some context of the question also.

Is it in a specific section that could be used to hint as to the working out process? -that would help identify whether there is information missing (ie, question is wrong)

or if, in context there are assumptions that can be made that we wouldn’t make out of context

or whether variables should be introduced as suggested.

**GabiAbi84**)It would help to know some context of the question also.

Is it in a specific section that could be used to hint as to the working out process? -that would help identify whether there is information missing (ie, question is wrong)

or if, in context there are assumptions that can be made that we wouldn’t make out of context

or whether variables should be introduced as suggested.

[Although as I remarked recently - it does seem that there are an *awful* lot of badly written questions going around at the minute. I'm really glad I'm no longer of an age where I need to do aptitude tests, because the current ones seem pretty moronic].

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