A triangle has corners A, B, and C located at #(8 ,7 )#, #(4 ,5 )#, and #(6 , 7 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
May 29, 2018

#"The coordinates of D (x, y) are (6.4, 6.2)."#
#"length=0.89"#

Explanation:

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  • Let's find the equation of the g line that passes through the triangle A and the B corner.

  • If the coordinates of two points of a line are known, then the equation of that line is written as follows.

  • # (y_2-y_1)/(x_2-x_1)=(y-y_2)/(x-x_2) #

  • #A(8,7) , B(4,5) , x_1=8 , y_1=7 , x_2=4 , y_2=5#

  • # (5-7)/(4-8)=(y-5)/(x-4)#

  • #(-2)/(-4)=(y-5)/(x-4)#

  • #1/2=(y-5)/(x-4)#

  • #x-4=2y-10#

  • #x-2y=-6 " (1) equation of line g"#

  • #y=1/2 x+3 #

  • If the equation is written in the form y = m x + n, m gives the slope. m=#1/2#

  • The altitude passing through the corner C will be perpendicular to line g.

  • Let D (x, y) be the intersection point.

  • multiplied by the slopes of two straight lines perpendicular to each other equal to -1.

  • #1/2 * m_f=-1 " , " m_f=-2#

  • #y-y_1=m(x-x_1#

  • #y-7=-2(x-6)" , "y-7=-2x+12#

  • #y+2x=19 " " (2)" the f line"#

  • Now we have two equations((1) and (2)).

  • (2) We multiply both sides of the equation by 2.

  • #2y+4x=38 " "(3)#

  • #x-2y=-6" "(1) #

  • Let's sum up the equations (1) and (3) we get .

  • 5x=32 , x=6.4

  • In equation (1) or (3) we write 6.4 instead of x

  • 6.4 -2y=-6 , -2y=-12.4 , y=6.2

  • The coordinates of D (x, y) are (6.4, 6.2).

  • length=#sqrt((6.4-6)^2+(6.2-7)^2)#

  • length=#sqrt((0.4)^2+(-0.8)^2)#

  • length=#sqrt((0.16)+(0.64))#

  • length=#0.89#