SOLUTION: Dear math tutor, A(7,2) and C(1,4) are two vertices of a square ABCD. Find the equation of the diagonal BD. Find the coordinates of B and D. with gratitude, nats

Algebra ->  Coordinate-system -> SOLUTION: Dear math tutor, A(7,2) and C(1,4) are two vertices of a square ABCD. Find the equation of the diagonal BD. Find the coordinates of B and D. with gratitude, nats      Log On


   

Question 870669: Dear math tutor,
A(7,2) and C(1,4) are two vertices of a square ABCD. Find the equation of the diagonal BD. Find the coordinates of B and D.
with gratitude,
nats
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!

Find the slope of the line connecting A and C.
m=%284-2%29%2F%281-7%29=2%2F-6=-1%2F3
Diagonal BD is perpendicular to diagonal AC so their slopes are negative reciprocals of each other.
m%5Bperp%5D%2A%28-1%2F3%29=-1
m%7Bperp%5D=3
Diagonal BD also goes through the midpoint of AC.
x%5Bm%5D=%287%2B1%29%2F2=8%2F2=4
y%5Bm%5D=%282%2B4%29%2F2=6%2F2=3
Use this point to find the line, BC, y=3x%2Bb
3=3%284%29%2Bb
b=-9
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y=3x-9

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The distance from the midpoint to points B and D is the same as the distance from the midpoint to A and C.
D%5E2=%287-4%29%5E2%2B%282-3%29%5E2
D%5E2=9%2B1
D%5E2=10
The distance from the midpoint to B and D is,
10=%28x-4%29%5E2%2B%28y-3%29%5E2
But you also know that B and D lie on the line y=3x-9
So substituting.
10=%28x-4%29%5E2%2B%283x-9-3%29%5E2
10=x%5E2-8x%2B16%2B9x%5E2-72x%2B144
10=10x%5E2-80x%2B160
10x%5E2-80x%2B150=0
x%5E2-8x%2B15=0
%28x-3%29%28x-5%29=0
Two solutions:
x-3=0
x=3
Then
y=3%283%29-9
y=0
and
x-5=0
x=5
Then,
y=3%285%29-9
y=6
B and D are (3,0) and (5,6)
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