ABCD is a parallelogram if P is a point on AC such that PC:PA=1:3 and area of triangle BPC = 16 cm² Find area of APD.

Given that ABCD is a parallelogram and P is a point on AC such that the ratio PC:PA = 1:3. Also, the area of triangle BPC is given as 16 cm². We need to find the area of triangle APD.

Step 1: Understanding the given ratio
We have the point P dividing AC in the ratio PC:PA = 1:3. This means:

If PC = x, then PA = 3x.
Therefore, the entire diagonal AC = PC + PA = x + 3x = 4x.
This implies that P divides AC in a 1:3 ratio.
Step 2: Area ratio based on the section
Since P divides AC in the ratio 1:3, it means that triangle APC is divided in the same proportion. That is:

Area of triangle BPC : Area of triangle APD is also 1:3.
Since area of triangle BPC is given as 16 cm², we set up the proportion:

Area of BPC / Area of APD = 1/3
16 / Area of APD = 1/3
Area of APD = 16 × 3 = 48 cm².

Final Answer:
The area of triangle APD is 48 cm².