My Math Forum Triangle possibilities

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 May 14th, 2013, 12:55 AM #1 Newbie   Joined: May 2013 Posts: 3 Thanks: 0 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?
 May 14th, 2013, 08:11 AM #2 Senior Member   Joined: Sep 2010 Posts: 221 Thanks: 20 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.
 May 14th, 2013, 09:02 AM #3 Global Moderator   Joined: Dec 2006 Posts: 18,250 Thanks: 1439 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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# possibility 9f triangle

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