ruben has two congruent wooden dowels. he cuts one dowel in two in order to have three pieces to make a…

Brief Explanations:

To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side (Triangle Inequality Theorem). Let the length of each congruent dowel be ( l ). After cutting one dowel into two pieces, say lengths ( a ) and ( b ) (so ( a + b = l )), the three pieces have lengths ( a ), ( b ), and ( l ). Now, check the triangle inequality: ( a + b = l ), but for a triangle, ( a + b ) must be greater than ( l ). Since ( a + b = l ) (not greater), the condition fails. Thus, Ruben can't form a triangle.

Answer:

To form a triangle, the sum of any two side - lengths must be greater than the third side (Triangle Inequality Theorem). Let the length of each congruent dowel be ( l ). When one dowel is cut into two pieces with lengths ( a ) and ( b ), we have ( a + b=l ). The three pieces of wood have lengths ( a ), ( b ), and ( l ). For the three lengths to form a triangle, ( a + b>l ) must hold. But since ( a + b = l ) (from the cut), this inequality is not satisfied. So, Ruben cannot make a triangle with the three pieces.