Question

Let O (0,0), P (3,4) and Q (6,0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR and OQR are equal in area. Then the coordinates of R are:
[a] $({\displaystyle \frac{4}{3}},3)$
[b] $(3,{\displaystyle \frac{2}{3}})$
[c] $(3,{\displaystyle \frac{4}{3}})$
[d] $({\displaystyle \frac{4}{3}},{\displaystyle \frac{2}{3}})$

Answer 1

Hint: Use the property that in a triangle ABC if AP is the median of a triangle ABC, the areas of triangles APB and APC are equal.
Use the property that the coordinates of the centroid of triangle ABC with $\text{A}\equiv (x_1,y_1),\text{B}\equiv (x_2,y_2)$ and $\text{C}\equiv (x_3,y_3)$ are given by $({\displaystyle \frac{x_1+x_2+x_3}{3}},{\displaystyle \frac{y_1+y_2+y_3}{3}})$ .

Complete step-by-step answer:

Given: Coordinates of point O, P and Q of a triangle OPQ. Areas of triangles OPR, PQR and OQR are equal
To determine: Coordinates of point R.
Claim: R is the centroid of the triangle OPQ
Property: If a point inside the triangle is such that two of the triangles formed by joining the point with the vertices of the triangle, then the side common to those two triangles is the median of the triangle.
Hence since the triangles OPR and PQR are equal, PR is a median of the triangle
By a similar argument OR and QR are also medians of the triangle
Hence R lies on all the three medians of the triangle OPR.
Since medians of a triangle are concurrent at the centroid, R is the centroid of the triangle OPR.
Now, we know that the coordinates of the centroid of triangle ABC with $\text{A}\equiv (x_1,y_1),\text{B}\equiv (x_2,y_2)$ and $\text{C}\equiv (x_3,y_3)$ are given by $({\displaystyle \frac{x_1+x_2+x_3}{3}},{\displaystyle \frac{y_1+y_2+y_3}{3}})$ .
Here $x_1=0,x_2=3,x_3=6,y_1=0,y_2=4$ and $y_3=0$
Hence $\text{R}\equiv ({\displaystyle \frac{0+3+6}{3}},{\displaystyle \frac{0+4+0}{3}})=(3,{\displaystyle \frac{4}{3}})$
Hence option [c] is correct.
Note: [1] In the above question, we have used the property that the median of a triangle divides the triangle into two triangles of equal area. It can be proved by constructing an altitude from the point where the median is drawn and using the formula for finding the area of a triangle.
[2] Although in the above question, we have proved that R is the centroid of the triangle, you need to remember the result.
“If cevians of a triangle divide the triangle into six triangles of equal area then the point of concurrence of the cevians is the centroid of the triangle”.