In an isosceles triangle, from the apex of the angle at the base to the lateral side, a median is drawn

In an isosceles triangle, from the apex of the angle at the base to the lateral side, a median is drawn, which divides the perimeter of the triangle into parts of 8 cm and 15 cm. Find the base of the trackball.

Let us denote the lengths (in centimeters) of the side (equal to each other) sides through a and the length (also in centimeters) of the base through x.
According to the condition of the assignment, a median is drawn from the vertex of the angle at the base to the lateral side, therefore, the perimeter of the triangle is divided into two parts: one part contains one side completely and half of the other side, that is, in formal form this part is equal to a + 0 , 5 * a = 1.5 * a; the other part contains the base completely and half of the side, that is, in formal form, this part is equal to x + 0.5 * a.
The description of the assignment does not specify which part of the perimeter is 8 cm and which is 15 cm.However, in any case, its perimeter is 8 cm + 15 cm = 23 cm, therefore, the half-perimeter is (23 cm): 2 = 11.5 see. Consider the theoretically possible two cases and check the results obtained (that is, the lengths of the sides of the triangle) for the existence of a triangle. For verification, we will use the following fact: in any triangle, the half-perimeter is larger than any of its sides.
First case. Let 1.5 * a = 8 cm and x + 0.5 * a = 15 cm.Then, a = (8 cm): 1.5 = 16/3 cm = 5⅓ cm.Consequently, x = 15 cm – 0 , 5 * a = 15 cm – 0.5 * (16/3) cm = 37/3 cm = 12⅓ cm. Since 11.5 cm <12⅓ cm, this case will not give real results (in this case, the triangle is not exists).
Second case. Let 1.5 * a = 15 cm and x + 0.5 * a = 8 cm.Then, a = (15 cm): 1.5 = 10 cm.Therefore, x = 8 cm – 0.5 * a = 8 cm – 0.5 * 10 cm = 3 cm.Since 11.5 cm> 10 cm and 11.5 cm> 3 cm, we can confidently assert that the length of the base of the triangle is 3 cm.
Answer: 3 cm.