Triangle Area #3

Geometry Level 2

The coordinates of the vertices of a triangle are given to be A = ( 1 , 7 ) , B = ( 4 , 5 ) , C = ( 10 , 12 ) A=(1,7), B= (4,5), C= (10,12) . Find the area of the triangle A B C ABC .


The answer is 16.5.

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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2 solutions

Area of a Polygon (Coordinate Geometry)

A = 1 2 [ x 1 x 2 x 3 . . . x n x 1 y 1 y 2 y 3 . . . y n y 1 ] A=\dfrac{1}{2} \begin{bmatrix} x_1 & x_2 & x_3 ...& x_n & x_1 \\ y_1 & y_2 & y_3 ...& y_n & y_1 \end{bmatrix}

Inputting the points and simplifying, we obtain

A = 1 2 [ 1 4 10 1 7 5 12 7 ] A=\dfrac{1}{2} \begin{bmatrix} 1 & 4 & 10 & 1 \\ 7 & 5 & 12 & 7 \end{bmatrix} A = 1 2 [ ( 1 × 5 ) + ( 4 × 12 ) + ( 10 × 7 ) ] − 1 2 [ ( 7 × 4 ) + ( 5 × 10 ) + ( 12 × 1 ) ] = 1 2 [ ( 5 + 48 + 70 ) − ( 28 + 50 + 12 ) ] A=\dfrac{1}{2}[(1\times5)+(4\times12)+(10\times7)]-\dfrac{1}{2}[(7\times4)+(5\times10)+(12\times1)]=\dfrac{1}{2}[(5+48+70)-(28+50+12)]

A = 1 2 ( 123 − 90 ) = 1 2 ( 33 ) = A=\dfrac{1}{2}(123-90)=\dfrac{1}{2}(33)= 16.5 \large\color{#3D99F6}\boxed{16.5} answer \color{#69047E}\boxed{\text{answer}}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

like mine itself but presented in a different way.→

Syed Hamza Khalid - 4 years, 1 month ago

You can try out https://brilliant.org/problems/triangle-area-1/ and https://brilliant.org/problems/triangle-area-2-2/

Syed Hamza Khalid - 4 years, 1 month ago

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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