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Question
PQR is an isosceles triangle . If two of its vertices are P (2 , 0) and Q (2 , 5) , find the coordinates of R if the length of each of the two equal sides is 3.
Solution
PQ = c
∴ PR = QR = 3 units
Let the coordinates of R be on ,
PR = `sqrt (("x" - 2)^2 + ("y" - 0)^2)`
`=> 3 = sqrt ("x"^2 + 4 - 4"x" + "y"^2)`
squaring both sides ,
`=> 9 = "x"^2 - 4"x" + "y"^2 + 4`
`=> "x"^2 - 4"x" + "y"^2 - 5 = 0`
`=> "x"^2 + "y"^2 - 4"x" = 5` .....(1)
QR = `sqrt (("x" - 2)^2 + ("y" - 5)^2)`
`=> 3 = sqrt ("x"^2 + 4 - 4"x" + "y"^2 + 25 - 10"y")`
⇒ 9 = x2 + y2 - 4x - 10y + 29
⇒ 0 = x2 + y2 - 4x - 10y + 29
From (1) 0 = 5 - 10y + 20
10 y = 25
y = `5/2`
`=> "x"^2 + 25/4 - 4"x" - 5 = 0`
`=> 4"x"^2 + 25 - 16"x" - 20 = 0`
`=> 4"x"^2 - 16"x" + 5 = 0`
D = (-16)2 - 4(4)(5)
= 256 - 80
= 176
`sqrt "d" = sqrt 176 = 4 sqrt 11`
x = `(16 +- 4 sqrt 11)/(2 xx 4)`
= `(4 +- 4 sqrt 11)/2`
= `2 + sqrt 11/2 , 2 - sqrt 11/2`
The coordinates of R are `(2 - sqrt 11/2 , 5/2)` or `(2 + sqrt 11/2 , 5/2)`
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