In TRIANGLE ABC points P,Q,R arethe points lies inside triangle in such a way that Ap= PR, BQ=QP &QR=RC n area of triangle PQR =100 then find the Area of ABC
To find the area of triangle ABC, we can use the concept of triangle similarity.
Given that AP = PR, BQ = QP, and QR = RC, we can conclude that triangle APR is similar to triangle PQR (by the Side-Side-Side similarity criterion) and triangle BQR is similar to triangle PQR (by the Side-Side-Side similarity criterion).
Let's denote the length of AP (or PR) as x and the length of QR (or RC) as y. Since triangle APR is similar to triangle PQR, the ratio of corresponding sides is equal:
AP/PQ = PR/QR
x/(x + y) = (x + y)/y
By cross-multiplication, we have:
xy = (x + y)²
xy = x² + 2xy + y²
x² + y² = xy
Similarly, in triangle BQR, we have:
BQ/PQ = QR/PR
(x + y)/(x + 2y) = y/x
By cross-multiplication, we have:
xy = (x + y)²
x² + y² = 2xy
By substituting x² + y² = xy (from triangle APR) into x² + y² = 2xy (from triangle BQR), we get:
xy = 2xy
This equation implies that xy = 0, which is not possible since the area of a triangle cannot be zero.
Therefore, there is no solution for the given scenario where AP = PR, BQ = QP, and QR = RC.