According to this theorem ” two triangles having the same base and lies between the same parallel lines are equal in area”
In the above figure;
ABC and PBC are the two triangles having same base BC.
Also, both the triangles lies between the same parallel lines.
To Prove:
Area ( ABC ) = Area ( PBC )
Construction:
Draw line CD parallel to AB
Draw line CR parallel to BP
Proof:
After the construction we now have two parallelogram ABCD and PBCR.
Note that both the parallelogram have the same base and lie between the same parallel lines.
So according to parallelogram area theorem we can write;
Area ( ABCD ) = Area ( PBCR ) —> eq (1)
Now consider triangle ABC and parallelogram ABCD.
Both the figures have same base and lie between same parallel lines.
According to parallelogram – triangle theorem, we can write:
Area (ABC) = 1/2 Area (ABCD)
2 * Area (ABC) = Area (ABCD) —> eq (2)
Similar is the case of triangle PBC and parallelogram PBCR
Area (PBC) = 1/2 Area (PBCR)
2 * Area (PBC) = Area (PBCR) —> eq (3)
Applying eq (2) and (3) on equation (1), we get;
Area ( ABCD ) = Area ( PBCR )
2 * Area (ABC) = 2 * Area (PBC)
Area (ABC) = Area (PBC)
Hence Proved