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Section Formula

Let Aa⃗ , ...

Question

Let $A (\to a)$ , $B (\to b), C (\to c)$ be the vertices of the triangle $A B C$ and let $D E F$ be the mid points of the sides $B C, C A, A B$ respectively. If $P$ divides the median $A D$ in the ratio $2 : 1$ then the position vector of $P$ is

0

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\to a + \to b + \to c

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\frac{\to a + \to b + \to c}{3}

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\frac{2 \to a + \to b + \to c}{3}

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Solution

The correct option is C $\frac{\to a + \to b + \to c}{3}$
Given :
$A (\to a), B (\to b), C (\to c)$ are the vertices of the triangle $A B C$ .
$D$ is the midpoint of $B C$
$⟹ D = (\frac{\to b + \to c}{2})$
$E$ is the midpoint of $C A$
$⟹ E = (\frac{\to a + \to c}{2})$
$F$ is the midpoint of $A B$
$⟹ F = (\frac{\to a + \to b}{2})$
Now, $A D, B E$ and $C F$ are the medians of triangle $A B C$
All three medians intersects at point $P$ .
$∵ P$ divides $A D$ in the ratio $2 : 1$ and other medians also intersect at $P$ .
$⟹ P$ divides all three medians in the ratio $2 : 1$
$⟹ P$ is the centroid of the triangle $A B C$ .
$⟹ P = \frac{\to a + \to b + \to c}{3}$ ...... By def $^{n}$ of centroid.
Hence, option C is correct.

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Similar questions

$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to

Q. Statement-

1

: If vertices of a triangle are

A (\to a)

B (\to b)

C (\to c)

then length of altitude through

A

\frac{| \to a \times \to b + \to b \times \to c + \to c \times \to a |}{| \to b - \to c |}

Statement-

2

: Area of triangle is

Δ = \frac{1}{2}

(Base) (Height)

= \frac{1}{2} | \to A B \times \to A C |

Q. Let

\to a, \to b, \to c

be three vectors in the

x y z

space such that

\to a \times \to b = \to b \times \to c = \to c \times \to a \neq \to 0

.
If

A, B, C

are points with position vectors

\to a, \to b, \to c

respectively, then the number of possible positions of the centroid of triangle

A B C

is.

Q. If

\to a, \to b, \to c

are positive vectors of vertices

A, B, C

Δ A B C

. If

\to r

is position vector of a point

P

such that

(| \to b - \to c | + | \to c - \to a | + | \to a - \to b |) \to r = | \to b - \to c | \to a + | \to c - \to a | \to b + ∣ ∣ \to a - \to b ∣ ∣ \to c

then the point

P

always

Q. Assertion :Let

A (\to a), B (\to b)

and

C (\to c)

be three points such that

\to a = 2 ˆ i + ˆ k, \to b = 3 ˆ i - ˆ j + 3 ˆ k

and

\to c = - ˆ i + 7 ˆ j - 5 ˆ k

, then

O A B C

is a tetrahedron. Reason: Let

A (\to a), B (\to b)

and

C (\to c)

be three points such that vectors

\to a, \to b

and

\to c

are non-coplanar, then

O A B C

is a tetrahedron, where

O

is the origin.

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Let A(→a) , B(→b),C(→c) be the vertices of the triangle ABC and let DEF be the mid points of the sides BC,CA,AB respectively. If P divides the median AD in the ratio 2:1 then the position vector of P is

Let $A (\to a)$ , $B (\to b), C (\to c)$ be the vertices of the triangle $A B C$ and let $D E F$ be the mid points of the sides $B C, C A, A B$ respectively. If $P$ divides the median $A D$ in the ratio $2 : 1$ then the position vector of $P$ is