In triangle PQR, base QR is divided at x such that

QX =1/2 QR

To Prove-ar(PQX) =1/3 ar(PQR)

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Solution

Given: PQR is a triangle
X divides the base QR in the ratio of 2:1 i.e., QX= 1/2 XR
rtp: ar triangle PQX =1/3ar triangle PQR
construction: join PX
draw an altitude PY for triangle PQR
proof: area of triangle PQX = 1/2 PYQX
area of triangle PXR = 1/2 * PY * XR
= 1/2 * PY * 2QX
= 2(1/2 * PY * QX)
= 2( area of triangle PQX)
we know that, area of triangle PQR = area of triangle PQX + area of triangle PXR
= 3( area of triangle PQX)
area of triangle PQX = 1/3(area of triangle PQR )
hence proved

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