Question

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?

Answer 1

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

Explanation:

• 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`