Answer to Question #21620 in Geometry for korena

Question

4.Cake.Carl has a piece of cake in the shape of an isosceles triangle with angles 26 ° ,77 ° , and 77 °. He wanted to divide it into two equal parts, so he cut it through the middle of the 26 ° angle to the midpoint of the opposite side. He says that because he is dividing it at the midpoint of a side, the two pieces are congruent. Is this enough information? Explain.

Answer 1

Carl is correct, he has cut the large piece into 2 equal smaller pieces. What he drew was not only an angle bisector, but also a median. When you draw a median of a triangle, it does indeed cut the other side into 2 equal segments. Let's label our large triangle ABC, with A being the vertex with the 26 degree angle. Let's call the midpoint of the segment opposite vertex A, point D. Now, the 2 triangles that are formed are triangle ADB, and triangle ADC. They are congruent because AB is congruent to AC since it is an isosceles triangle. BD is congruent to CD since D is the midpoint of BC.