CBSE Class 10-Mathematics: Chapter –7 Coordinate Geometry Part 27
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Question 23:
Let and be the vertices of
(i) The median from A meets BC at D. Find the coordinates of the point D.
(ii) Find the coordinates of the point P on AD such that .
(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that .
(iv) What do you observe?
(Note: The point which is common to all the three medians is called centroid and this point divides each median in the ratio )
(v) If A are the vertices of , find the coordinates of the centroid of the triangle.
The Vertices of ABC
Answer:
Let and be the vertices of
(i) Since AD is the median of .
D is the mid-point of BC.
Its coordinates are
(ii) Since P divides AD in the ratio
Its coordinates are
(iii) Since BE is the median of .
E is the mid-point of AD.
Its coordinates are
Since Q divides BE in the ratio .
Its coordinates are
Since CF is the median of .
F is the mid-point of AB.
Its coordinates are
Since R divides CF in the ratio .
Its coordinates are
(iv) We observe that the points P, Q and R coincide, i.e., the medians AD, BE and CF are concurrent at the point . This point is known as the centroid of the triangle.
(v) According to the question, D, E, and F are the mid-points of BC, CA, and AB, respectively.
Coordinates of a point dividing AD in the ratio are
The coordinates of E are
The coordinates of a point dividing BE in the ratio are
Similarly, the coordinates of a point dividing CF in the ratio are
Thus, the point is common to AD, BE and CF and divides them in the ratio .
The median of a triangle are concurrent, and the coordinates of the centroid are