In a ΔABC, P and Q are respectively the midpoints of AB and BC and R is the midpoint of  line segment AP. Then if ar(ΔRQC) = k ar(ΔABC). Then k equals

The problem is asking for the ratio of the area of triangle RQC to the area of triangle ABC.

Step 1: Understand the problem In triangle ABC, P and Q are the midpoints of AB and BC respectively. This means that AP = PB and BQ = QC. R is the midpoint of AP, which means that AR = RP.

Step 2: Apply the properties of triangles The area of a triangle is given by 1/2 * base * height. Since P and Q are midpoints, triangle APQ is similar to triangle ABC, and their areas are in the ratio of the squares of their corresponding sides.

Step 3: Calculate the ratio of the areas Since AP = 1/2 AB and BQ = 1/2 BC, the area of triangle APQ is (1/2)^2 = 1/4 the area of triangle ABC.

Step 4: Apply the properties of triangles again R is the midpoint of AP, so triangle ARQ is similar to triangle APQ, and their areas are in the ratio of the squares of their corresponding sides. Since AR = 1/2 AP, the area of triangle ARQ is (1/2)^2 = 1/4 the area of triangle APQ.

Step 5: Calculate the final ratio Since triangle RQC is the same as triangle ARQ, the area of triangle RQC is 1/4 the area of triangle APQ, which is 1/4 the area of triangle ABC. Therefore, the area of triangle RQC is (1/4)^2 = 1/16 the area of triangle ABC.

So, k = 1/16.