If the medians of a ΔABC intersect at G, show that : ar (ΔAGB) = ar (ΔAGC) = ar (ΔBGC) = ar (ΔABC).

Question

KnowledgeBoat · Accepted Answer

In ΔABC, Median AD divides ΔABC into two Δs of equal area. ∴ ar (ΔABD) = ar (ΔACD) ....(1) In ΔGBC, Median GD divides ΔGBC into two Δs of equal area. ∴ ar (ΔGBD) = ar (ΔGCD) ....(2) Subtracting equation (2) from equation (1), we get : ar (ΔABD) − ar (ΔGBD) = ar (ΔACD) − ar (ΔGCD) ar (ΔAGB) = ar (ΔAGC) ...........(3) In ΔABC, Median BE divides ΔABC into two Δs of equal area. ∴ ar (ΔABE) = ar (ΔBCE) ....(4) In ΔGAC, Median GE divides ΔGAC into two Δs of equal area. ∴ ar (ΔGEA) = ar (ΔGEC) ....(5) Subtracting equation (5) from equation (4), we get : ar (ΔABE) − ar (ΔGEA) = ar (ΔBCE) − ar (ΔGEC) ar (ΔAGB) = ar (ΔBGC) ...........(6) From equation (3) and (6), we get : ar (ΔAGB) = ar (ΔAGC) = ar (ΔBGC) From figure, ar (ΔAGB) + ar (ΔAGC) + ar (ΔBGC) = ar (ΔABC) 3ar (ΔAGB) = ar (ΔABC) ar (ΔAGB) = × ar (ΔABC) Hence, proved that ar (ΔAGB) = ar (ΔAGC) = ar (ΔBGC) = ar (ΔABC).

Mathematics

If the medians of a ΔABC intersect at G, show that : ar (ΔAGB) = ar (ΔAGC) = ar (ΔBGC) = ar (ΔABC).

Theorems on Area

Related Questions

In the given figure, a point D is taken on side BC of ΔABC and AD is produced to E, making DE = AD. Show that :
ar (ΔBEC) = ar (ΔABC).

In the given figure, AD is a median of ΔABC and P is a point on AC such that :
ar (ΔADP) : ar (ΔABD) = 2 : 3.
Find :
(i) AP : PC
(ii) ar (ΔPDC) : ar (ΔABC)

Mathematics

If the medians of a ΔABC intersect at G, show that : ar (ΔAGB) = ar (ΔAGC) = ar (ΔBGC) = ar (ΔABC).

Theorems on Area

Related Questions

In the given figure, a point D is taken on side BC of ΔABC and AD is produced to E, making DE = AD. Show that :ar (ΔBEC) = ar (ΔABC).

In the given figure, AD is a median of ΔABC and P is a point on AC such that :ar (ΔADP) : ar (ΔABD) = 2 : 3.Find :(i) AP : PC(ii) ar (ΔPDC) : ar (ΔABC)

In the given figure, a point D is taken on side BC of ΔABC and AD is produced to E, making DE = AD. Show that :
ar (ΔBEC) = ar (ΔABC).

In the given figure, AD is a median of ΔABC and P is a point on AC such that :
ar (ΔADP) : ar (ΔABD) = 2 : 3.
Find :
(i) AP : PC
(ii) ar (ΔPDC) : ar (ΔABC)