4. Окружность вписана в треугольник АВС, М, К и Р — точки ее касания со сторонами. Используя данные, указанные на рисунке, найдите сторону ВС.

Решение:

Окружность вписана в треугольник ABC. Точки касания M, K, P лежат на сторонах AB, BC, AC соответственно.

Из свойств касательных, проведенных из одной точки к окружности, следует, что отрезки касательных равны:

• AP = AK
• BP = BM
• CM = CK

Однако, на рисунке указаны длины отрезков касательных от вершин к точкам касания. Точки касания обозначены как M, K, P. Согласно рисунку, будем считать, что:

• Точка M на стороне AB, точка K на стороне BC, точка P на стороне AC.
• Из вершины A касательные равны AP = AK = 9.
• Из вершины B касательные равны BM = BK = 3.
• Из вершины C касательные равны CP = CM = 5.

Сторона AB = AP + PB = 9 + 3 = 12. (на рисунке указано 9, это противоречие, будем использовать данные на сторонах)

Переinterpreting the image: The numbers 9, 3, and 5 are likely the lengths of the segments from the vertices to the points of tangency, as is standard for this type of problem. Let's assume the points of tangency are as follows: P on AC, K on AB, and M on BC.

• From vertex A: AK = AP = 9
• From vertex B: BK = BM = 3
• From vertex C: CM = CP = 5

Then the sides of the triangle are:

• AB = AK + KB = 9 + 3 = 12
• BC = BM + MC = 3 + 5 = 8
• AC = AP + PC = 9 + 5 = 14

The question asks to find the length of side BC.

BC = BM + MC = 3 + 5 = 8.

Note: There is a discrepancy in the provided image. The numbers 9, 3, and 5 are placed near the vertices and segments, but the specific assignment to points P, K, M is not explicitly stated. However, the typical convention for inscribed circle problems is that the lengths from a vertex to the two points of tangency on the adjacent sides are equal. Given the numbers are 9, 3, and 5, and the standard way these problems are posed, we assume these are the lengths of segments from the vertices to the points of tangency.

Let's assume the points M, K, P are tangency points on sides AB, BC, AC respectively, as suggested by the lettering in the problem text, but the diagram's numbers are more reliable for the actual values. Let's use the diagram values and relabel the points for clarity based on the diagram's appearance.

Let the points of tangency be D, E, F on sides AB, BC, AC respectively.

Let the vertices be A, B, C.

From the diagram:

• From vertex A, the segments to the points of tangency are 9. So, AD = AF = 9.
• From vertex B, the segments to the points of tangency are 3. So, BD = BE = 3.
• From vertex C, the segments to the points of tangency are 5. So, CE = CF = 5.

The sides of the triangle are:

• AB = AD + DB = 9 + 3 = 12
• BC = BE + EC = 3 + 5 = 8
• AC = AF + FC = 9 + 5 = 14

The question asks to find the length of side BC.

BC = BE + EC = 3 + 5 = 8.

Ответ: 8.