Question

The vertices of ΔABC are (−2, 1), (5, 4)  and (2, −3)  respectively. Find the area of the triangle and the length of the altitude through A.

Answer 1

GIVEN: The vertices of triangle ABC are A (−2, 1) and B (5, 4) and C (2, −3)

TO FIND: The area of triangle ABC and length if the altitude through A

PROOF: We know area of triangle formed by three points (x_1,y₁),(x_2,y₂)and (x_3,y₃)is given 

 by Δ `=1/2[x_1(y_1-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)  ` 

NOW AREA OF ΔABC 

Taking three point A(-2,1) and B(5,4) and C(2,-3)  

Area (ΔABC)`=1/2[{-8-15+2}-{5+8+6}]` 

`=1/2[{-21}-{19}]` 

`=1/2[{-40}]` 

`=1/2(40)` 

`=20` 

WE HAVE 

`BC=sqrt((5-2)^2+(4+3)^2)`  

`BC=sqrt((3)^2+(7)^2)`

`BC =sqrt(9+49 )` 

`BC=sqrt58 `

NOW  

 Area  (ΔABC)`=1/2xxBCxx`ength of altitude though A

20`=1/2xxsqrt 58xx `lenght of altitude through A 

 Lenght of altitude through A`=40/sqrt58`