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Choose the statement that is true/false.

ΔABC is right angled at C. If A=(1,2), and the circumcenter of ΔABC is at (6,2) and length of side AC is 8 units, then the possible coordinates of C are (7.4, 2.8).

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True

False

answer is B.

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Detailed Solution

AB is the hypotenuse, BC is the base and AC is the perpendicular.
Let two points be

(x_{1}, y_{1})

and

(x_{2}, y_{2})

and their mid point be

(x_{m}, y_{m})

(x_{m}, y_{m}) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Given that A=(1, 2) and the circumcenter P has the coordinates (6,2). Let coordinates of B are (p,q).

(6, 2) = (\frac{1 + p}{2}, \frac{2 + q}{2})

\Rightarrow

p = 11 and q = 2
Let C be (a, b).
AC =

\sqrt{{(a - 1)}^{2} + {(b - 2)}^{2}} = 8

AB =

\sqrt{{(11 - 1)}^{2} + {(2 - 2)}^{2}} = 10

BC =

\sqrt{{(a - 11)}^{2} + {(b - 2)}^{2}}

Since ABC is a right angle triangle.

{AB}^{2} = {AC}^{2} + {BC}^{2}

\Rightarrow 10^{2} = 8^{2} + {(a - 11)}^{2} + {(b - 2)}^{2}

\Rightarrow {(a - 11)}^{2} + {(b - 2)}^{2} = 36

\Rightarrow {(b - 2)}^{2} = 36 - {(a - 11)}^{2} = 64 - {(a - 1)}^{2}

\Rightarrow - a^{2} - 121 + 22 a = 28 - a^{2} - 1 + 2 a

\Rightarrow 20 a = 148

\Rightarrow

a = 7.4 And b = -2.8
Hence, The given statement is false.

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