A triangle has corners A, B, and C located at (5 ,5 ), (3 ,9 ), and (4 , 1 ), respectively. What are the endpoints and length of the altitude going through corner C?

2 Answers
Jul 27, 2018

The end points of altitude CD is (4,1) and (6.4,2.2)
Length of altitude CD ~~ 2.68 unit

Explanation:

A(5,5) , B(3,9) , C(4,1)

Let CD be the altitude going through C touches D on line

AB. C and D are the endpoints of altitude CD; CD is

perpendicular on AB. Slope of AB= m_1= (y_2-y_1)/(x_2-x_1)

=(9-5)/(3-5) = 4/(-2) =-2 :. Slope of

CD=m_2= -1/m_1= 1/2

Equation of line AB is y - y_1 = m_1(x-x_1) or

y- 5 = -2(x-5) or 2 x+y = 15 ; (1)

Equation of line CD is y - y_3 = m_2(x-x_3) or

y- 1 = 1/2(x-4) or 2 y-2= x-4 or 2 y - x =-2 ;(2)

Sollving equation (1) and (2) we get

the co-ordinates of D(x_4,y_4). Mutiplying equation (2) by 2

we get -2x+4y=-4 ; (3) , adding equation (1) from

equation (3) we get 5 y=11 or y=11/5=2.2

:. x=(15-y)/2=(15-2.2)/2=6.4 :. D is (6.4,2.2,).

The end points of altitude CD is (4,1) and (6.4,2.2)

Length of altitude CD is

CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2) or

CD = sqrt((4-6.4)^2+(1-2.2)^2)= sqrt 7.2 ~~ 2.68 unit [Ans]

Jul 27, 2018

The endpoints of altitude are :C(4,1) and N(32/5,11/5).
The length of altitude is :CN=sqrt(36/5)

Explanation:

Let triangleABC " be the triangle with corners at"

A(5,5), B(3,9) and C(4,1)

Let bar(AL) , bar(BM) and bar(CN) be the altitudes of sides bar(BC) ,bar(AC) and bar(AB) respectively.

Let (x,y) be the intersection of three altitudes

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Slope of bar(AB) =(5-9)/(5-3)=-4/2=-2

bar(AB)_|_bar(CN)=>slope of bar(CN)=1/2 ,

bar(CN) passes through C(4,1)

:.The equn. of bar(CN) is :y-1=1/2(x-4)

=>2y-2=x-4

=>x-2y=2

i.e. color(red)(x=2y+2.....to (1)

Now, Slope of bar(AB) =-2 and bar(AB) passes through
A(5,5)

So, eqn. of bar(AB) is: y-5=-2(x-5)

=>y-5=-2x+10

=>color(red)(y=15-2x...to(2)

From (1)and (2)

y=15-2(2y+2)=15-4y-4=11-4y

=>y+4y=11=>5y=11=>color(blue)(y=11/5

From (1) ,

x=2(11/5)+2=>color(blue)(x=32/5

=>N(32/5,11/5) and C(4,1)

Using Distance formula,

CN=sqrt((32/5-4)^2+(11/5-1)^2)=sqrt(144/25+36/25)
:.CN=sqrt(180/25)

:.CN=sqrt(36/5)~~2.68