What is the altitude of an isosceles triangle with congruent side lengths of 8 cm and a vertex angle of 120 degrees? What is the length of the non-congruent side?

To find the altitude of the isosceles triangle, we first need to find the length of the base (the non-congruent side).

Step 1: Find the length of the base The isosceles triangle can be split into two 30-60-90 right triangles by drawing a line from the vertex angle to the midpoint of the base. The sides of a 30-60-90 triangle are in the ratio 1:√3:2. Since the hypotenuse (the congruent side of the isosceles triangle) is 8 cm, the base of each right triangle (which is half the base of the isosceles triangle) is 4 cm. Therefore, the length of the base of the isosceles triangle is 2*4 = 8 cm.

Step 2: Find the altitude The altitude of the isosceles triangle is the length of the side opposite the 60-degree angle in the 30-60-90 triangle. This is √3 times the length of the side opposite the 30-degree angle. Since the side opposite the 30-degree angle is 4 cm, the altitude is 4√3 cm.

So, the altitude of the isosceles triangle is 4√3 cm and the length of the non-congruent side is 8 cm.