Question

In ΔPQR, PS is a median. T is the mid-point of SR and M is the mid-point of PT. Prove that: ΔPMR = ${\displaystyle \frac{1}{8}\Delta \text{PQR}}$.

Answer 1

Area(ΔPQR) = area(ΔPQS) + area(ΔPSR) ....(i)
Since PS is the median of ΔPQR and median divides a triangle into two triangles of equal area.
Therefore, area(ΔPQS) = area(ΔPSR)
Substituting in (i)
Area(ΔPQR) = area(ΔPSR) + area(ΔPSR)
Area(ΔPQR) = 2area(ΔPSR) .........(iii)
Area(ΔPSR) = area(ΔPST) + area(ΔPTR) .....(iv)
Since PT is the median of ΔPSR and median divides a triangle into triangles of equal area.
Therefore, area(ΔPST) = area(ΔPTR)  .....(v)
Substituting in (v)
Area(ΔPSR) = 2area(ΔPTR) ........(vi)
Substituting in (iii)
Area(ΔPQR) = 2 x 2area(ΔPTR)
Area(ΔPQR) = 4area(ΔPTR) .........(vii)
Area(ΔPQR) = area(ΔPMR) + area(ΔMTR) .....(viii)
Since MR is the median of ΔPTR and median divides a triangle into two triangles of equal area.
Therefore, area(ΔPMR) = area(ΔMTR)  ....(ix)
Substituting in (viii)
Area(ΔPQR) = 4 x 2area(ΔPMR)
Area(ΔPQR) = 8 x area(ΔPMR)

area(ΔPMR) = ${\displaystyle \frac{1}{8}\text{area(ΔPQR)}}$.