Question

A triangle has corners A, B, and C located at `(4 ,8 )`, `(7 ,4 )`, and `(5 ,3 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

End points of the altitude are `(5,3) & (6.76,4.32)`
Length of altitude `2.2` unit**

Explanation:

Let CD be the altitude drawn from C perpendicular to side AB touches at D of side AB.
Slope of line AB is `(4-8)/(7-4) = -4/3 :.`slope of line CD is `-1/(-4/3) =3/4` Equation of line CD is `y-3=3/4(x-5) or 4y-12=3x-15 or 4y-3x= -3; (1)` Equation of line AB is `y-8 = -4/3(x-4) or 3y-24 =-4x +16 or 3y+4x = 40; (2)`
Now solving equation (1) & equation (2) we find the co-ordinate of poin D. `12y-9x = -9` (1)x3 (3)
`12y+16x = 160` (2)x4 (4)
subtracting (3) from (4) we get `25x=169 or x =169/25=6.76 :. y =(40-(4*6.76))/3 =4.32` End points of the altitude are `(5,3) & (6.76,4.32)`
Length of altitude `= sqrt((5-6.76)^2+(3-4.32)^2) =2.2`[Ans]