Question

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Answer 1

Question -

Two vertices of ∆ ABC are A ( 6, 4 ) & B ( -2, 2 ) .

If the Centroid is G ( 3, 4 ) , find the third vertex of the triangle and also find the area of ∆ ABC and also find the area of ∆ABC

[ Zen Paper 3 - 2020 ]

Solution -

In the above Question , the following information is given -

Two vertices of ∆ ABC are A ( 6, 4 ) & B ( -2, 2 ) .

The Centroid is G ( 3, 4 ) .

Let the third vertex be C ( x, y ) .

Thus , the x coordinates of C is x .

The y coordinate of C is y .

Now ,

Centroid -

If the three vertices of a triangle have the following coordinates , ( x¹ , y¹ ) , ( x², y² ) , ( x³ , y ³ ) .

Centroid -

Centroid is [ x¹ + x² + x³ / 3 ] , [ y¹ + y² + y³ / 3 ]

Note that here , x¹ , x² etc don't represent any exponents , but I have named as for convinience .

So ,

In this case -

Three vertices of ∆ ABC are A ( 6, 4 ) & B ( -2, 2 ) . and C ( x, y )

A = ( 6, 4 )

B = ( -2, 2 )

C = ( 5, 12 )

Thus -

x¹ = 6

x² = -2

x³ = 5

y¹ = 4

y² = 2

y³ = 12

Centroid -

=> [ 6 - 2 + x / 3 ] , [ 4 + 2 + y / 3 ]

=> [ 4 + x / 3 ] , [ 6 + y / 3 ]

But the given centroid is ( 3, 4 )

Therefore -

[ 4 + x / 3 ] = 3

=> 4 + x = 9

=> x = 5

[ 6 + y / 3 ] = 4

=> 6 + y = 12

=> y = 6

Thus , coordinates of point C

=> C ( 5, 6 )

Now, we have to find the area of ∆ABC

A = ( 6, 4 )

B = ( -2, 2 )

C = ( 5, 6 )

Thus -

x¹ = 6

x² = -2

x³ = 5

y¹ = 4

y² = 2

y³ = 6

Area of ∆ABC

=> ( 1 / 2 ) × | x¹ y² + x² y ³ + x³ y¹ - y¹ x² - y² x³ - y³ x¹ |

Substuting the given Values -

=> ( 1 / 2 ) × | 12 - 12 + 20 + 8 - 10 - 36 |

=> ( 1 / 2 ) × | - 18 |

=> ( 1 / 2 ) × 18

=> 9 unit² .

Answer -

C = ( 5, 6 )

Area of ∆ ABC = 9 unit²

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