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# An obtuse triangle is a triangle that has an interior angle

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An obtuse triangle is a triangle that has an interior angle [#permalink]

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24 Jul 2006, 06:34
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An obtuse triangle is a triangle that has an interior angle whose measure is greater than 90 degrees. John has drawn all obtuse triangles with an area of 20 square inches whose angle measures in degrees are all multiples of 10. How many triangles has he drawn?

(A) 13 (B) 14 (C) 15 (D) 16 (E) more than 16
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24 Jul 2006, 07:10
kevincan wrote:
An obtuse triangle is a triangle that has an interior angle whose measure is greater than 90 degrees. John has drawn all obtuse triangles with an area of 20 square inches whose angle measures in degrees are all multiples of 10. How many triangles has he drawn?

(A) 13 (B) 14 (C) 15 (D) 16 (E) more than 16

First of all, i think the data of the area is a distraction since with any 3 fixed angle measures, we can adjust the sides to get the required area.
Since one angle ( the obstute one) can be 100,110,120.....170 ( 180 is not counted coz the triangle will become a line then)
+ angle is 100, we have the other two angles be (10,70) , (20,60 ) , (30,50), (40,40) ..the order is not important here.
+ 110: ( 10,60) , (20,50), (30,40)
+ 120: (10,50) , (20,40), (30,30)
+130: ( 10,40) , (20,30)
+140: (10,30), (20,20)
+150: (10,20)
+160: (10,10)
170: such triangle does not exist.

==> we have 16 such triangles
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24 Jul 2006, 09:14
We don't need to consider area here, 'cuz we can draw to match the area of triangle with given interior angles.

180=10(a+b+c)
18 = a+b+c (where a,b,c are integers)
let c>9

if c=10
a+b = 8
a can be 1,2,3,4,(5,6,7) (redundancies)
4 possible outcomes.
if c=11
a+b = 7
a can be 1,2,3,(4,5,6)
3 possible outcomes.
if c=12
a+b = 6
a can be 1,2,3,(4,5)
3 possible outcomes
.....
Total possible outcomes = (4+3+3+2+2+1+1) = 16

Thus, D must be it.
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24 Jul 2006, 09:24
I think E. You can have infinite combinations.

Area = 1/2 * base * Height.

You can vary the base and height to get any number of triangles. The obtuse information is just a distraction from the real problem.
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24 Jul 2006, 09:28
Got 16 too. (Assuming we don't consider a triangle with 180,0,0 angles as a trinagle.)
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24 Jul 2006, 11:47
laxieqv wrote:
kevincan wrote:
An obtuse triangle is a triangle that has an interior angle whose measure is greater than 90 degrees. John has drawn all obtuse triangles with an area of 20 square inches whose angle measures in degrees are all multiples of 10. How many triangles has he drawn?

(A) 13 (B) 14 (C) 15 (D) 16 (E) more than 16

First of all, i think the data of the area is a distraction since with any 3 fixed angle measures, we can adjust the sides to get the required area.
Since one angle ( the obstute one) can be 100,110,120.....170 ( 180 is not counted coz the triangle will become a line then)
+ angle is 100, we have the other two angles be (10,70) , (20,60 ) , (30,50), (40,40) ..the order is not important here.
+ 110: ( 10,60) , (20,50), (30,40)
+ 120: (10,50) , (20,40), (30,30)
+130: ( 10,40) , (20,30)
+140: (10,30), (20,20)
+150: (10,20)
+160: (10,10)
170: such triangle does not exist.

==> we have 16 such triangles

Great! the area wasn't a distraction. A better wording would be "no triangle is similar to any other" I 'm glad nearly everybody caught it.
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25 Jul 2006, 01:33
D 16 triangles

With obtuse angle 100 - 4 traingles (70,10) (60,20) (50,30) (40,40)
With obtuse angle 110 - 3 triangles
With obtuse angle 120 - 3 triangles
With obtuse angle 130 - 2 triangles
With obtuse angle 140 - 2 triangles
With obtuse angle 150 - 1 triangle
With obtuse angle 160 - 1 triangle
With obtuse angle 170 - 0
Total = 16
25 Jul 2006, 01:33
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