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An obtuse triangle is a triangle that has an interior angle [#permalink]
24 Jul 2006, 05:34

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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?

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 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

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.