Given the diagram, calculate the perimeter of triangle ABC.

Analysis:

• The image shows a right-angled triangle ABC with an inscribed circle.
• The vertices are labeled A, B, and C.
• The points where the circle touches the sides are labeled K, O, and M.
• The side lengths are given as:
• AK = 8
• KB = 8
• BM = 10
• MC = 10
• CL = 6
• LA = 6

Note: The labels K, O, M appear to be for the points of tangency. Let's assume the lengths are segment lengths from vertices to points of tangency.

In a triangle with an inscribed circle, the lengths of the two tangent segments from a vertex to the circle are equal.

• From vertex A: The segments are to points of tangency on AB and AC. Let's denote these points as P on AB and Q on AC. AP = AQ.
• From vertex B: The segments are to points of tangency on BA and BC. Let's denote these points as P on AB and R on BC. BP = BR.
• From vertex C: The segments are to points of tangency on CA and CB. Let's denote these points as Q on AC and R on BC. CQ = CR.

Looking at the diagram and labels, it seems the labels 8, 10, and 6 are indeed the lengths of these tangent segments.

• From vertex A, the tangent segments appear to be AK and a segment on AC (let's call it AL). So, AK = AL = 8.
• From vertex B, the tangent segments appear to be BK and BM. So, BK = BM = 10.
• From vertex C, the tangent segments appear to be CM and CL. So, CM = CL = 6.

However, the diagram labels are slightly different. Let's reinterpret based on the labels shown:

• Figure 23: Triangle ABC with inscribed circle.
• Vertex A is connected to points E and F. AE = 4, AF = 4.
• Vertex B is connected to points E and M. BE = 10, BM = 10.
• Vertex C is connected to points M and F. CM = 6, CF = 6.

Based on this interpretation:

• Side AB = AE + EB = 4 + 10 = 14
• Side BC = BM + MC = 10 + 6 = 16
• Side AC = AF + FC = 4 + 6 = 10

The question asks for P_ΔABC, which is the perimeter of triangle ABC.

Perimeter = AB + BC + AC

Perimeter = 14 + 16 + 10 = 40

Ответ: 40