Amzad Sadik wrote:A triangle has side lengths of a, b, and c centimeters. Does each angle in the triangle measure less than 90 degrees?
(1) The 3 semicircles whose diameters are the sides of the triangle have areas that are equal to 3 cm^2, 4 cm^2, and 6 cm^2, respectively.
(2) c < a + b < c + 2
$$?\,\,\,\,\,\,:\,\,\,\,\,\,{a^2} < {b^2} + {c^2}\,\,\,\,\,AND\,\,\,\,\,{b^2} < {a^2} + {c^2}\,\,\,\,\,AND\,\,\,\,\,{c^2} < {a^2} + {b^2}\,\,\,\,\,\,\,\,\,\,\,\left[ {\,\Delta = \left( {a,b,c} \right)\,\,\,{\rm{side}}\,\,{\rm{lengths}}\,} \right]$$
$$\left( 1 \right)\,\,{\rm{lengths}}\,\,{\rm{of}}\,\,{\rm{sides}}\,\,{\rm{are}}\,\,\left( {{\rm{uniquely}}} \right)\,\,{\rm{known}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$
(We call this a "blind decision", because we know the answer is unique although we won´t need to discover if it is a "Yes" or if it is a "No"!)
$$\left( 2 \right)\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a;b;c} \right) = \left( {1;2;2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {a;b;c} \right) = \left( {3 \cdot {1 \over 2};4 \cdot {1 \over 2};5 \cdot {1 \over 2}} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left[ {{c^2} = {a^2} + {b^2}} \right] \hfill \cr} \right.$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
