Tag: coordinates, points and lines

Questions Related to coordinates, points and lines

The area of $\displaystyle \Delta $ whose vertices are (a,b+c),(b,c+a),(c,a+b) will be -

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. 0

  2. a+b+c

  3. ab+bc+ac

  4. None of these

Show answer
Correct Option: A
Explanation:

Area of $\displaystyle \Delta $
$\displaystyle =\frac { 1 }{ 2 } \left[ a\left( c+a \right) -b\left( b+c \right) +b\left( a+b \right) -c\left( c+a \right) +c\left( b+c \right) -a\left( a+b \right)  \right] $
$\displaystyle =\frac { 1 }{ 2 } \left[ ac+{ a }^{ 2 }-{ b }^{ 2 }-bc+ab+{ b }^{ 2 }-{ c }^{ 2 }-ac+bc+{ c }^{ 2 }-{ a }^{ 2 }-ab \right] $
$\displaystyle =\frac { 1 }{ 2 } \times 0$
$\displaystyle =0$

If $A,B,C$ are the vertices of a triangle whose position vectors are $\vec { a } ,\vec { b } ,\vec { c } $ and $G$ is the centroid of the $\triangle ABC$, then $\overrightarrow { GA } +\overrightarrow { GB } +\overrightarrow { GC } $ is

mathematics and statistics coordinates, points and lines what is meant by the equation of a straight line or of a curve introduction to slope introduction to straight lines

  1. $\vec { 0 } $

  2. $\vec { A } +\vec { B } +\vec { C } $

  3. $\cfrac { a+b+c }{ 3 } $

  4. $\cfrac { a-b-c }{ 3 } $

Show answer
Correct Option: A
Explanation:

Given the position vectors of vertices $A,B$ and $C$ of the triangle $ABC$ are $\vec { a } ,\vec { b } $ and $\vec { c } $


ie $\overrightarrow { OA } =\vec { a } \quad \overrightarrow { OB } =\vec { b } \quad \overrightarrow { OC } =\vec { c } $

$\therefore$ Centroid of triangle $(G)=\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 } $

Now $\overrightarrow { GA } +\overrightarrow { GB } +\overrightarrow { GC } $

$=\left( \overrightarrow { OA } -\overrightarrow { OG }  \right) +\left( \overrightarrow { OB } -\overrightarrow { OG }  \right) +\left( \overrightarrow { OC } -\overrightarrow { OG }  \right) $

$=\left( \vec { a } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) +\left( \vec { b } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) +\left( \vec { c } -\cfrac { \vec { a } +\vec { b } +\vec { c }  }{ 3 }  \right) \quad $

$=\cfrac { 1 }{ 3 } \left( 3\vec { a } -\vec { a } -\vec { b } -\vec { c } +3\vec { b } -\vec { a } -\vec { b } -\vec { c } +3\vec { c } -\vec { a } -\vec { b } -\vec { c }  \right) $

$=\cfrac { 1 }{ 3 } \left[ \vec { 0 }  \right] =\vec { 0 } $