Let A(4,2),B(6,5) and C(1,4) be the vertices of ΔABC.(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 AP:PD=2:1(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ:QE=2:1 and CR:RF=2:1.(iv) What do yo observe?[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2:1.](v) If A(x1,y1),B(x2,y2) and C(x3,y3) are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Question

Let A(4,2),B(6,5) and C(1,4) be the vertices of ΔABC.(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 AP:PD=2:1(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ:QE=2:1 and CR:RF=2:1.(iv) What do yo observe?[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2:1.](v) If  A(x1​,y1​),B(x2​,y2​)  and  C(x3​,y3​)  are the vertices of ΔABC, find the coordinates of the centroid of the triangle.

Filo · Accepted Answer

i)

Median is the line joining the midpoint of one side of a triangle to the opposite vertex. So, the coordinates of

D

would be

(\frac{6 + 1}{2}, \frac{5 + 4}{2})

::

(\frac{7}{2}, \frac{9}{2})

ii)

P

divides

A D

in the ratio

2 : 1

.

A (x_{1}, y_{1}) = (4, 2), D (x_{2}, y_{2}) = (\frac{7}{2}, \frac{9}{2})

m : n = 2 : 1

Using section formula, we get the coordinates of

P

.

P (x, y) = (\frac{n x _{1} + m x _{2}}{m + n}, \frac{n y _{1} + m y _{2}}{m + n})

= ⎝ ⎛ \frac{1 \cdot 4 + 2 \cdot \frac{7}{2}}{2 + 1}, \frac{1 \cdot 2 + 2 \cdot \frac{9}{2}}{2 + 1} ⎠ ⎞

= (\frac{11}{3}, \frac{11}{3})

iii)

Coordinates of

E

will be

(\frac{5}{2}, 3)

and the coordinates of

F = (5, \frac{7}{2})

.

Coordinates of

Q = (\frac{n x _{1} + m x _{2}}{m + n}, \frac{n y _{1} + m y _{2}}{m + n})

= ⎝ ⎛ \frac{1 \cdot 6 + 2 \cdot \frac{5}{2}}{2 + 1}, \frac{1 \cdot 5 + 2 \cdot 3}{2 + 1} ⎠ ⎞

= (\frac{11}{3}, \frac{11}{3})

Coordinates of

R = (\frac{n x _{1} + m x _{2}}{m + n}, \frac{n y _{1} + m y _{2}}{m + n})

= ⎝ ⎛ \frac{1 \cdot 1 + 2 \cdot 5}{2 + 1}, \frac{1 \cdot 4 + 2 \cdot \frac{7}{2}}{2 + 1} ⎠ ⎞

= (\frac{11}{3}, \frac{11}{3})

iv)

The coordinates of

P, Q

and

R

are the same which is

(\frac{11}{3}, \frac{11}{3})

.

This point is called the centroid, denoted by

G

.

v)

Centroid of triangle

A BC = (\frac{x _{1} + x _{2} + x _{3}}{3}, \frac{y _{1} + y _{2} + y _{3}}{3})

Question Text	Let $A (4, 2), B (6, 5)$ and $C (1, 4)$ be the vertices of $Δ A BC$ . (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 $A D$ such that $A P : P D = 2 : 1$ (iii) Find the coordinates of points $Q$ and $R$ on medians $BE$ and $CF$ respectively such that $BQ : QE = 2 : 1$ and $CR : RF = 2 : 1$ . (iv) What do yo observe? [Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio $2 : 1$ .] (v) If $A (x_{1}, y_{1}), B (x_{2}, y_{2})$ and $C (x_{3}, y_{3})$ are the vertices of $Δ A BC$ , find the coordinates of the centroid of the triangle.
Answer Type	Text solution:1
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Text solutionVerified

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