SOLUTION: Triangle ABC has vertices A (-5, 4), B (1, -2) and C (3, 6), (a)Write the equation of the line of which AB is a segment. (b)Write the equation of the line of which the altitude t

Algebra ->  Algebra  -> Triangles -> SOLUTION: Triangle ABC has vertices A (-5, 4), B (1, -2) and C (3, 6), (a)Write the equation of the line of which AB is a segment. (b)Write the equation of the line of which the altitude t      Log On

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Question 332266: Triangle ABC has vertices A (-5, 4), B (1, -2) and C (3, 6),
(a)Write the equation of the line of which AB is a segment.
(b)Write the equation of the line of which the altitude to line AC is a segment.
(c)Write the equation of the perpendicular bisector of line AC.
(d)Find the perimeter of triangle ABC.
Found 2 solutions by D'Leeter, Edwin McCravy:
Answer by D'Leeter(4) About Me  (Show Source):
You can put this solution on YOUR website!


(a). Use the two-point form of an equation of a line:



where and are the coordinates of points A and C.

(b). Step 1. Use the



portion of the two-point form to determine the slope of the line containing segment AB.

Step 2: Calculate the negative reciprocal of the slope determined in Step 1 because:



Step 3: Use the point-slope form of an equation of a line:



where are the coordinates of point C and is the slope calculated in (b) Step 2.

(c) Step 1: Use the mid-point formulas:

and



where and are the coordinates of points A and C to calculate the midpoint of segment AC.

Use the slope calculated in (b) Step 2 and the midpoint calculated in (c) Step 1 with the point-slope form to derive the equation of the perpendicular bi-sector of AC.

(d) Step 1: Use the distance formula 3 times:







where , , and are the coordinates of points A, B, and C .

Step 2: Sum the three results.


John

My calculator said it, I believe it, that settles it

Answer by Edwin McCravy(8999) About Me  (Show Source):
You can put this solution on YOUR website!

Triangle ABC has vertices A (-5, 4), B (1, -2) and C (3, 6),

That triangle is:

(a)Write the equation of the line of which AB is a segment.

That's the green line below:





y-y%5B1%5D=m%28x-x%5B1%5D%29

y-%284%29=%28-1%29%28x-%28-5%29%29

y-4=-%28x%2B5%29

y-4=-x-5

y=-x-1

(b)Write the equation of the line of which the altitude to line AC is a segment.

That's the blue line below:



Since it is an altitude, it is perpendicular to AC.

First we find the slope of AC:



The blue line is perpendicular to AC so its slope is
the reciprocal of 1%2F4 with the opposite sign, so the
blue line's slope is -4 and it passes through B(1,-2)

y-y%5B1%5D=m%28x-x%5B1%5D%29

y-%28-2%29=%28-4%29%28x-%281%29%29

y%2B2=-4%28x-1%29

y%2B2=-4x%2B4

y=-4x%2B2

(c) Write the equation of the perpendicular bisector of line AC.

We find the midpoint of AC, using the midpoint formula:

Midpoint = (%28x%5B1%5D%2Bx%5B2%5D%29%2F2,%28y%5B1%5D%2By%5B2%5D%29%2F2+%29) = (%28%28-5%29%2B%283%29%29%2F2,%284%2B6%29%2F2+%29) = (%28-2%29%2F2,%2810%29%2F2+%29) = M(-1,5)

Now we find the equation of the red line through M(-1,5), it has the same 
slope as the altitude since both are perpendicular to the line AC.



The red line's slope is also -4 but it passes through M(-1,5)

y-y%5B1%5D=m%28x-x%5B1%5D%29

y-%285%29=%28-4%29%28x-%28-1%29%29

y-5=-4%28x%2B1%29

y-5=-4x-4

y=-4x%2B1

(d)Find the perimeter of triangle ABC.

We use the distance formula to find the lengths of all three sides.

d=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29







So the perimeter = AB + BC + AC = 6sqrt%282%29%2B2sqrt%2817%29%2B2sqrt%2817%29=6sqrt%282%29%2B+4sqrt%2817%29

Edwin