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May 13th, 2013, 11:55 PM   #1
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Triangle possibilities

Let x, y, ? be positive real numbers with ? < ? . We want to construct a triangle in which two sides have lengths x and y and one angle has measure ?. Up to congruence, how many such triangles are there? For each statement below, write whether it is TRUE or FALSE.
a) For some values of x, y, ? there is no such triangle.
b) For some values of x, y, ? there is exactly one such triangle.
c) For some values of x, y, ? there are exactly two such triangles.
d) For some values of x, y, ? there are exactly three such triangles

I get why a) is false but can somebody explain to me, or prove why b), c), d), are true?
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May 14th, 2013, 07:11 AM   #2
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Re: Triangle possibilities

With any given angle there are always two possibilities: if x>y the triangle is possible when
1) the angle is between x and y;
2) the angle is between y and the third side.
For definite values of x, y, and angle <90, all three possibilities exist.
So a) and b) are false; c) and d) are true.
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May 14th, 2013, 08:02 AM   #3
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That analysis is incorrect.

(a) This is false, as a triangle exists with sides of length x and y, and the given angle in between them.

(b) This is true. If x = y and the given angle is ?/3, the triangle is equilateral.

(c) This is true. If x = 3, y = 4 and the angle is ?/2, the third side is either 5 or ?7.

(d) This is true. Example: x = 1 + ?3, y = ?2 and angle is ?/6.
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