State whether the given statement is true or false.If a

ΔABC
,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of

ΔABG
,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of and

ΔACD
and is the same.

Question

State whether the given statement is true or false.If a

ΔABC
 ,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of

ΔABG
 ,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of and

ΔACD
  and  is the same.

Accepted Answer

We have been given,If the

ΔABC
 has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of

ΔABG
  and

ΔACD
  is the same.Let the x1, x2,x3, y1,y2 and y3 are coordinates of vertices of ΔABC, then the centroid of a triangle is,

x
        1
       
       +
        x
        2
       
       +
        x
        3

3
     
     ,

y
        1
       
       +
        y
        2
       
       +
        y
        3

3

Using the formula for centroid.Substituting x1=3, x2=6, x3=9, y1=2, y2=4, and y3=3, in the formula,

G≡

3+6+9
          3
         
         ,
          
           2+4+3
          3

⇒G≡(6,3)

Now, calculating the area of the triangles.ar(ΔABG)=12x1y2-y3+x2y3-y1+x3y1-y2Substituting x1=3, x2=6, x3=6, y1=2, y2=4, and y3=3, in the formula,arΔABG=1234-3+63-2+62-4arΔABG=12|3+6-12|arΔABG=12|-3|arΔABG=32Area of triangle ACG,ar(ΔACG)=12x1y2-y3+x2y3-y1+x3y1-y2Substituting x1=3, x2=9, x3=6, y1=2, y2=3, and y3=3, in the formula,arΔACG=1233-3+93-2+62-3arΔACG=12|0+9-6|arΔACG=12|3|arΔACG=32arΔACG=arΔABG=32 sq. unitTherefore, the given statement is true.Hence, option 1 is true.

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State whether the given statement is true or false.If a ΔABC ,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of ΔABG ,has the coordinates of points A, B, and C as (3,2),(6,4) and (9,3) respectively, then the areas of and ΔACD and is the same.