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Question

Show that A(2,3),B(4,5) and C(3,2) can be the vertices of a rectangle. Find the coordinates of the fourth vertex.

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Solution

Given: A=(2,3);B=(4,5);C=(3,2)
Slope of AB=5342=1
Slope of BC=2534=31=3
Slope of AC=2332=1
Slope of AB× Slope of AC=1
ABAC
Let D=(x,y)
Slope of AC=Slope of BD
3223=5y4x
5y=x4x+y9=0
Also, Slope of BD× Slope of DC=1
5y4x×y2x3=1
(5y)(y2)=(3x)(4x)
5y10y2+2y=123x4x+x2
y27y+10=x2+7x12
Puttig y=9x
(9x)27(9x)+10+x27x+12=0
81+x218x63+7x+10+x27x+10=0
2x218x+40=0
x29x+20=0x=5 or x=4
If x=5,y=4
If x=4,y=5
(4,5) is already a vertex (B)
D=(5,4)
Four vertices are A(2,3);B(4,5);C(3,2);D(5,4)

1166684_1184568_ans_311a46e3ee2d47aabcf125019089f06a.JPG

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