
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]
[b]
[c]
[d]
Answer
530.1k+ views
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 and are given by .
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 and are given by .
Here and
Hence
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”.
Use the property that the coordinates of the centroid of triangle ABC with
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
Here
Hence
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”.
Recently Updated Pages
Master Class 11 Business Studies: Engaging Questions & Answers for Success

Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 Computer Science: Engaging Questions & Answers for Success

Master Class 11 Maths: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Trending doubts
Which one is a true fish A Jellyfish B Starfish C Dogfish class 11 biology CBSE

The flightless birds Rhea Kiwi and Emu respectively class 11 biology CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

1 Quintal is equal to a 110 kg b 10 kg c 100kg d 1000 class 11 physics CBSE
