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Author Topic: Math Problem (Not Kite Related)  (Read 1931 times)
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chilese
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« on: January 19, 2013, 01:37 PM »

Found a SAT problem that I couldn't figure out.

Without using references I am stymied.

Any of you want to help?

I know that (a+c) = (d+e), but can't get beyond that.

I'm guessing that B is the answer, but don't know why.



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« Reply #1 on: January 19, 2013, 05:39 PM »

It's been over 25 years, but I'm game.

What do we know:
a+b+c = b+d+e=180 degrees.
Therefore a+c = d+e

Forget it. I don't remember.
« Last Edit: January 19, 2013, 06:44 PM by Tmadz » Logged
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« Reply #2 on: January 19, 2013, 07:19 PM »

This one is a bit of a stumper.  Since there are no measurements whatsoever you really have nothing but visual cues to go on and maybe a few simple rules. THis is not a math problem but a logic problem.

THe only things we can know for sure is that a+b+c=180 degrees  as should b+d+e. We can assume that a, b and c are not equal due to the lean of the triangle and that A and C are not equal.  By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles. Now since B is the same for both we can surmise that d+e=a+c.

Now IF c=d and d=e then c=e and since we know that c+b+a=180 and d+e+b=180 then we can also surmise that a is equal to c,d and e.  Therefore since answer E does not allow for a to be equal to any of them the answer is "B".  If a and e are equal then a+b=e+b therefore d+b=c+b.  B is the only answer that logically allows for all angles to be considered in the equation.
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« Reply #3 on: January 20, 2013, 06:39 AM »

I asked my brother, who's rather good with this sort of thing. Here's what he said:
Quote
There are theorems that the angle between a tangent and a cord is half the subtended arc, as is the inscribed angle. So a=e, c=d.
Proof: Draw a triangle with base db and opposite side the center of the circle, noting that it is an isosceles triangle with central angle the angle of the arc bd and the other angles equal to the complement of a (since a radius and a tangent meet in a right angle). The central angle is thus 180-2comp(a)=180-2(90-a)=2a.
The other (that e is half the angle of the inscribed arc) is a standard theorem, and looks a bit harder to prove.
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« Reply #4 on: January 20, 2013, 06:45 AM »

I'm guessing that B is the answer, but don't know why.

B is the correct answer.  But I can't tell you why.
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« Reply #5 on: January 20, 2013, 07:05 PM »

Tangent chord is half the central angle.
inscribed angle is also half the central angle, and therefore equal.

These are geometric rules

I did the first half, the 2nd is half is done the same way

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« Reply #6 on: January 21, 2013, 07:27 AM »

By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles.

You happen to exclude wrong answers and leave the correct answer, but this logic is just faulty. The question does not say the answer will contain "pairs." The set of five possible answers contains "pairs."
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« Reply #7 on: January 21, 2013, 01:09 PM »

Well, I had forgotten whatever they had tried to force into me at school about angles of chords, so this is how I worked it out....

Draw lines from the vertices of the triangles to the centre of the circle.  This splits the triangle up into three isoceles triangles (triangles with two equal sides and angles).  Label the equal angles of the upper isoceles triangle "p", the equal angles of the isoceles triangle to the right "q" and the equal angles of the isoceles triangle to the left "r". 

Regarding the original triangle, because the interior angles of a triangle add up to 180 degrees:  180 degrees = 2( p + q + r) , or p = 90 degrees - q - r.

Because the radius at the bottom is perpendicular to the tangent, r = 90 degrees - a, and q = 90 degrees - c.  Substituting this into the equation for p gives p = 90 degrees - (90 degrees - c) - (90 degrees - a) = c+a - 90 degrees

So d = p + r = c+a - 90 degrees + 90 degrees - a = c
e = p + q = c+a - 90 degrees + 90 degrees - c = a


...This is "kite related" for me, in that I have tried and tried to do tricks by trying to copy the actions of others, but usually failed at that.  The way my mind works, I have had to learn what little I have learnt by trying to figure out what is going on - how the kite can fly.
Hey, ho.   Bet the USA SATs don't work it out in the above way....
« Last Edit: January 21, 2013, 01:12 PM by damp_weather » Logged
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« Reply #8 on: January 21, 2013, 10:01 PM »

....
Hey, ho.   Bet the USA SATs don't work it out in the above way....

My opinion has always been it doesn't matter which road you take if you arrive at your destination ... I never did too well in math either, the right answer never seemed to satisfy the teacher because the convoluted road I took to get there always confused them  Sad   I think mine was more scenic though   Cheesy



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« Reply #9 on: January 21, 2013, 10:23 PM »

By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles.

You happen to exclude wrong answers and leave the correct answer, but this logic is just faulty. The question does not say the answer will contain "pairs." The set of five possible answers contains "pairs."
Actually no only B and D contain pairs, the other contain "a" pair.  Most, if not all multiple choice math questions are worded so you can pretty much eliminate 60% of the answers right off.  Once you know the most likely answers you can work backwards like I did to see which one is ultimately correct.
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chilese
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« Reply #10 on: January 21, 2013, 10:32 PM »

I read the pairs in "both" interpretations.

I ended up thinking the answer should be as general as possible,
so I didn't exclude the single pair answers.

I did use the property of mirror symmetry to exclude a few of the
potential answers as the "correct" answer would have to include
mirror symmetry in how the pairs were listed.
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« Reply #11 on: January 22, 2013, 09:12 AM »

By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles.
You happen to exclude wrong answers and leave the correct answer, but this logic is just faulty. The question does not say the answer will contain "pairs." The set of five possible answers contains "pairs."
Actually no only B and D contain pairs, the other contain "a" pair.  Most, if not all multiple choice math questions are worded so you can pretty much eliminate 60% of the answers right off.  Once you know the most likely answers you can work backwards like I did to see which one is ultimately correct.
I know we're both just repeating our positions here, but...
"Following pairs" means there are multiple to choose from. It does not necessarily mean the answer contains pairs. I'll admit you can parse it that way, but only because the verb is not one that indicates the number of the noun. (And the noun itself, "which", is no help, either.) However, the answer is not required to be one that contains multiple pairs.
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Ca Ike
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« Reply #12 on: January 22, 2013, 05:43 PM »

By the wording of the question you can eliminate A,C and d as possible answers so it has to be B or D since they are the only choices with "pairs" of angles.
You happen to exclude wrong answers and leave the correct answer, but this logic is just faulty. The question does not say the answer will contain "pairs." The set of five possible answers contains "pairs."
Actually no only B and D contain pairs, the other contain "a" pair.  Most, if not all multiple choice math questions are worded so you can pretty much eliminate 60% of the answers right off.  Once you know the most likely answers you can work backwards like I did to see which one is ultimately correct.
I know we're both just repeating our positions here, but...
"Following pairs" means there are multiple to choose from. It does not necessarily mean the answer contains pairs. I'll admit you can parse it that way, but only because the verb is not one that indicates the number of the noun. (And the noun itself, "which", is no help, either.) However, the answer is not required to be one that contains multiple pairs.
Your right, but from having a mother, who is also a teacher, I have seen a lot of standardized tests.  From my experience it's common for them to be worded to indicate what answers can be eliminated or if any can be.  Saying "which answer" or "which of these answers" if there is only one right or "which answers" if more than one. On similar geometry/calculous questions if its only one set of angles they are looking for it will usually say "which of these angles are the same" or "which pair of angles".  It doesn't always hold true but I'd say a good 95% of the time the question will clue you in to the most likely answer.  Even More so in the last few years they seem to be making it easier to guess correctly by how they word the questions.
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« Reply #13 on: January 26, 2013, 04:55 PM »

Huhm, I'll have to ponder this some. I think it will hinge on a geometry rule where an exterior angle for an oblique triangle (a or c) is equal to the sum of the opposite two interior angles.

But it reminds me of one I did solve:
sin(x)\n =y

By simple cancelation becomes:
y = si(x) = six =6 !!!

I wrote on the test if the prof would give me credit for a good try. He wrote a giant NO across it, stating there was no solution. But in a way I think he got a kick out of it anyway.

Albert
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