A triangle has corners A, B, and C located at $(3, 1)$ $(3 ,1 )$ , $(6, 4)$ $(6 ,4 )$ , and $(9, 8)$ $(9 ,8 )$ , respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-02-02T13:42:35

First find the line $A B$ $AB$
Then find the line $l$ $l$ perpendicular to $A B$ $AB$ (the line of altitude) such that it intersects with point $C$ $C$
Then find the point $D$ $D$ at which $l$ $l$ intersects with $A B$ $AB$
Then use the distance formula with the points $C$ $C$ and $D$ $D$ to get the distance of the altitude.

See if you can do it step by step. If not, keep on reading.

Step 1

Find the equation for $A B$ $AB$

$y - y_{1} = m (x - x_{1})$ $y-y_1=m(x-x_1)$ (point slope formula)

$m = \frac{4 - 1}{6 - 3} = 1$ $m = (4-1)/(6-3) = 1$ (finding the slope using the points $A$ $A$ and $B$ $B$ )

$y - 1 = x - 3$ $y-1=x-3$ (plugging $m$ $m$ and A back into the equation, you could choose A or B and it would be the same)

Segment $A B$ $AB$ lies on the line $y = x - 2$ $y=x-2$

Step 2

Find line $l$ $l$ that is perpendicular to $A B$ $AB$ and intersects with $C$ $C$ .

$y = m x + b$ $y = mx + b$ (start with an empty line)

$y = - 1 x + b$ $y = -1x + b$ (negative reciprocal slope of $A B$ $AB$ for perpendicular line)

$8 = - 9 + b$ $8 = -9 + b$ (substitute in point $C$ $C$ to find a line that intersects with $C$ $C$ )

$b = 17$ $b = 17$ (solve)

Now that we have $b$ $b$ we can list the equation.

$l$ $l$ , our altitude, has the equation $y = - x + 17$ $y = -x+17$

Step 3

Find point $D$ $D$ where our altitude $l$ $l$ and base $A B$ $AB$ intersect

$y = - x + 17$ $y = -x+17$ ( $l$ $l$ )
$y = x - 2$ $y=x-2$ ( $A B$ $AB$ )

So $- x + 17 = x - 2$ $-x+17=x-2$
$19 = 2 x$ $19=2x$
$x = 9.5$ $x = 9.5$
$y = 7.5$ $y = 7.5$ (by plugging $x$ $x$ back into one of the equations)

So #D = (9.5,7.5)

Step 4

Find the distance of $C D$ $CD$

$\sqrt{{(9 - 9.5)}^{2} + {(8 - 7.5)}^{2}}$ $sqrt((9-9.5)^2+(8-7.5)^2)$

$= \frac{\sqrt{2}}{2}$ $= sqrt(2)/2$

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A triangle has corners A, B, and C located at (3 ,1 )(3,1)(3 ,1 ), (6 ,4 )(6,4)(6 ,4 ), and (9 ,8 )(9,8)(9 ,8 ), respectively. What are the endpoints and length of the altitude going through corner C?