Find the coordinates of the point P on AD such that AP : PD $=$ 2 : 1

\frac{x_{1} + x_{2} + 2 x_{3}}{3}, \frac{y_{1} + y_{2} + 2 y_{3}}{3}

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\frac{2 x_{1} + 2 x_{2} + x_{3}}{3}, \frac{2 y_{1} + 2 y_{2} + y_{3}}{3}

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x_{1} + x_{2} + x_{3}, y_{1} + y_{2} + y_{3}

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\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}

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Solution

The correct option is D $\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}$
$T h e p o i n t s A (x_{1}, y_{1}), B (x_{2}, y_{2}) a n d C (x_{3} y_{3}) a r e t h e v e r t i c e s o f Δ A B C . ∴ T h e m e d i a n A D f r o m A w i l l m e e t B C a t D w h i c h i s t h e m i d p o i n t o f B C . S o, b y s e c t i o n f o r m u l a f o r m i d p o i n t, t h e c o - o r d i n a t e s o f D i s D (\frac{x_{2} + x_{3}}{2}, \frac{y_{2} + y_{3}}{2}) = (x_{4}, y_{4}) . N o w P d i v i d e s A D i n t h e r a t i o m : n = 2 : 1. ∴ T h e c o - o r d i n a t e s o f P (x, y) a r e, b y t h e s e c t i o n f o r m u l a, x = \frac{n x_{1} + n x_{4}}{m + n} = \frac{\frac{x_{2} + x_{3}}{2} \times 2 + x_{1} \times 1}{2 + 1} = \frac{x_{1} + x_{2} + x_{3}}{3} a n d y = \frac{n y_{1} + n y_{4}}{m + n} = \frac{\frac{y_{2} + y_{3}}{2} \times 2 + y_{1} \times 1}{2 + 1} = \frac{y_{1} + y_{2} + y_{3}}{3} ∴ P (x, y) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}) . A n s - P (x, y) = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}) .$

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Find the coordinates of the point P on AD such that AP : PD = 2 : 1

Find the coordinates of the point P on AD such that AP : PD $=$ 2 : 1