Question

A triangle has corners A, B, and C located at `(2 ,5 )`, `(7 ,4 )`, and `(6 ,1 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

End points of altitude are `(6,1) and (6.6,4.1)`
Length of altitude is `3.16 (2dp)` unit

Explanation:

Slope of the line AB is `m_1=(y_2-y_1)/(x_2-x_1) = (4-5)/(7-2) = -1/5`
Let CD be the altitude from C perpendicular on AB meets at D.
Slope of altitude CD is `m_2=-1/(-1/5)=5`[condition of perpendicularity is `m_1*m_2=-1`]
Equation of line AB is `y-5 = -1/5(x-2) or 5y +x =27 (1)`
Equation of line CD is `y-1 = 5(x-6) or y -5x = -29 (2 )`
`5y +x =27`(1)
`5y-25x= -145` (2)*5
Subtracting (2) from (1) we get `26x=172 or x= 86/13 = 6.62` ; `:. y= 5*6.62-29=4.1 :.`End points of altitude are `(6,1) and (6.6,4.1)`
Length of altitude is `sqrt((6.6-6)^2+(4.1-1)^2) =3.16`unit[Ans]