The vertices of a triangle are #(2, 1), (2,5),# and #(5, 1)#. What is the area of the triangle?

2 Answers
Jan 6, 2018

#6#

Explanation:

#"given the vertices of a triangle "#

#(x_1,y_1),(x_2,y_2)" and "(x_3,y_3)#

#"then the area (A) is given by"#

#A=1/2|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|#

#"let "(x_1,y_1)=(2,1),(x_2,y_2)=(2,5),(x_3,y_3)=(5,1)#

#A=1/2|2(5-1)+2(1-1)+5(1-5)|#

#color(white)(A)=1/2|8+0-20|#

#color(white)(A)=1/2|-12|=6#

Jan 6, 2018

#6#

Explanation:

this can also be calculated using determinant form

#for coordinates

#(x_1,y_1), (x_2,y_2),(x_3,y_3)#

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

in this case

#(2,1),(2,5),(5,1)#

#A=1/2|(2,1,1),(2,5,1),(5,1,1)|#

simplifying the determinant by elementary row operations

#R_3'=R_3-R_1#

#A=1/2|(2,1,1),(2,5,1),(3,0,0)|#

expand by #R_3# and taking the positive value

#A=||1/2(3|(1,1),(5,1)|-0+0)||#

#=||1/2xx3(1-5)||#

#=6#