A triangle has corners A, B, and C located at $(2, 9)$ $(2 ,9 )$ , $(1, 4)$ $(1 ,4 )$ , and $(6, 5)$ $(6 , 5 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

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A triangle has corners A, B, and C located at $(2, 9)$ $(2 ,9 )$ , $(1, 4)$ $(1 ,4 )$ , and $(6, 5)$ $(6 , 5 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-01-27T06:31:26

The endpoints are $(\frac{18}{13}, \frac{77}{13})$ $(18/13,77/13)$ and $(6, 5)$ $(6,5)$ and length of the altitude going through corner C is $4.707$ $4.707$

Explanation:

The slope of line $A B$ $AB$ is $\frac{4 - 9}{1 - 2} = \frac{- 5}{- 1} = 5$ $(4-9)/(1-2)=(-5)/(-1)=5$ and equation of $A B$ $AB$ is $(y - 4) = 5 (x - 1)$ $(y-4)=5(x-1)$

i.e. $5 x - y = 1$ $5x-y=1$ ................................(1)

Hence if $C D ⊥ A B$ $CD_|_AB$ , with $D$ $D$ on $A B$ $AB$ , slope of $C D$ $CD$ will be $\frac{- 1}{5} = - \frac{1}{5}$ $(-1)/5=-1/5$

and equation of $C D$ $CD$ is $(y - 5) = - \frac{1}{5} (x - 6)$ $(y-5)=-1/5(x-6)$

or $5 y - 25 = - x + 6$ $5y-25=-x+6$ i.e. $x + 5 y = 31$ $x+5y=31$ ................................(2)

Solving (1) and (2) gives us the coordinates of $D$ $D$ . For tis putting $y = 5 x - 1$ $y=5x-1$ from (1) in (2) gives

$x + 25 x - 5 = 31$ $x+25x-5=31$ or $26 x = 36$ $26x=36$ i.e. $x = \frac{36}{26} = \frac{18}{13}$ $x=36/26=18/13$

and hence $y = 5 \times \frac{18}{13} - 1 = \frac{77}{13}$ $y=5xx18/13-1=77/13$

and coordinated of $D$ $D$ are $(\frac{18}{13}, \frac{77}{13})$ $(18/13,77/13)$

and $C D = \sqrt{{(6 - \frac{18}{13})}^{2} + {(5 - \frac{77}{13})}^{2}}$ $CD=sqrt((6-18/13)^2+(5-77/13)^2)$

= $\sqrt{{(\frac{60}{13})}^{2} + {(- \frac{12}{13})}^{2}}$ $sqrt((60/13)^2+(-12/13)^2)$

= $\frac{1}{13} \sqrt{3600 + 144} = \frac{\sqrt{3744}}{13} = \frac{61.188}{13} = 4.707$ $1/13sqrt(3600+144)=sqrt3744/13=61.188/13=4.707$
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A triangle has corners A, B, and C located at $(2, 9)$ $(2 ,9 )$ , $(1, 4)$ $(1 ,4 )$ , and $(6, 5)$ $(6 , 5 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer

Explanation:

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A triangle has corners A, B, and C located at (2 ,9 )(2,9)(2 ,9 ), (1 ,4 )(1,4)(1 ,4 ), and (6 , 5 )(6,5)(6 , 5 ), respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer

Explanation:

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A triangle has corners A, B, and C located at $(2, 9)$ $(2 ,9 )$ , $(1, 4)$ $(1 ,4 )$ , and $(6, 5)$ $(6 , 5 )$ , respectively. What are the endpoints and length of the altitude going through corner C?