Geometry, question error...
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The attached image is a problem that has been bothering me for a little while... below is the solution but after reading it I think the diagram is a little off... particularly the location of the 90-x angle. Any one else see this or am I missing something?
Based on the provided diagram I am seeing that the sum of the interior angles of triangle DEC is:
180 = 90-x+90+angle E
180 = -x+180+angle E
0=-x+angle E
x = angle E
Since CD is opposite angle E, and angle E =x, CD must be equal to BC (=5).
Based on the diagram isn't this the correct solution?
Below is the provided solution....
[spoiler]Applying The Pythagorean Theorem to the right triangle ABC yields
BC2 + AC2 = AB2
52 + AC2 = 102 given that AB = 10 and BC = 5 (from the figure)
AC2 = 102 - 52 = 100 - 25 = 75
Square rooting yields AC = sqrt(75) = sqrt(25)sqrt(3) =5 sqrt(3.)
Hence, the sides opposite angles measuring x° (A in triangle ABC) and 90° - x° (B in triangle ABC) are in the ratio
5 : 5 sqrt(3) = 1 : sqrt(3)
Similarly, in triangle ECD, the ratio of the sides opposite the angles E (measuring x°) and D (measuring 90° - x°)
must also be 1 : sqrt(3)
Hence, we have
CD/EC = 1 : sqrt(3)
CD/(5 + 5) = 1 : sqrt(3)
CD = 10 /sqrt(3)
The answer is (C).[/spoiler]
- kevincanspain
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Those are two similar triangles, but they are not identical! Their hypotenuses have different lengths
Kevin Armstrong
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Thanks, I just had one of those "aha" moments but not in a good way. I don't know what led me to believe sides opposite equal angles in similar triangles were necessarily equal instead of simply proportional...
My common sense is working again!
My common sense is working again!
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excatly,i was wondering what formula is being used here at first instancegmatjedi wrote:the key to this problem is calculating length AC and recognizing that these are 30-60-90 triangles with lengths x-x[sq rt3]-2x
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Yeah, that shortcut is definitely valuable (I usually check for special right triangles right at the beginning) but just in case your senses leave you for whatever reason during the test, it's still worthwhile to understand the general approach using Pythagorean and the properties of similar triangles. IMHO.