A $△$ ABC where A (1, 2), B (4, 8) and C (6, 8) is reflected in the origin to $△$ $A_{1} B_{1} C_{1}$ . $△$ $A_{1} B_{1} C_{1}$ is then reflected in the x - axis to $△$ A₂, B₂, C₂. Write down the coordinates of A₂, B₂, C₂. Which single transformation could have provided the final co-ordinate of A₂, B₂, C₂?

$A_{2} (- 1, 2), B_{2} (- 4, 8), C_{2} (- 6, 8)$ ,
Transformation through x-axis

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$A_{2} (- 1, 2), B_{2} (- 4, 8), C_{2} (- 6, 8)$ ,
Transformation through y-axis

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$A_{2} (1, 2), B_{2} (4, 8), C_{2} (- 6, 8)$ ,
Transformation through origin

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None of these

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Solution

The correct option is B
$A_{2} (- 1, 2), B_{2} (- 4, 8), C_{2} (- 6, 8)$ ,
Transformation through y-axis

When a point is reflected in the origin, the sign of its ordinate and abscissa changes. Hence, the coordinates of $A_{1} B_{1} C_{1}$ are $A_{1} (- 1, - 2), B_{1} (- 4, - 8), C_{1} (- 6, - 8)$ .
When a point is reflected in the x axis, the sign of its ordinate changes. Hence, the coordinates are A₂ (-1, 2), B₂ (-4, 8), C₂ (-6, 8) .
A Single transformation about y-axis will provide final co-ordinates of A₂, B₂, C₂

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Similar questions

△ = ⎡ ⎢ ⎣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ⎤ ⎥ ⎦

A_{2}, B_{2}, C_{2}

a_{2}, b_{2}, c_{2}

a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2}

|\begin{matrix} - a (b^{2} + c^{2} - a^{2}) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & - b (c^{2} + a^{2} - b^{2}) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & - c (a^{2} + b^{2} - c^{2}) \end{matrix}| = a b c {(a^{2} + b^{2} + c^{2})}^{3}

a, b, c

a^{2} = b^{2} + c^{2} - 2 b c \sqrt{1 - a^{2}}, b^{2} = c^{2} + a^{2} - 2 c a \sqrt{1 - b^{2}}, c^{2} = a^{2} + b^{2} - 2 a b \sqrt{1 - c^{2}}

a = c \sqrt{1 - b^{2}} + b \sqrt{1 - c^{2}}

a + b + c = 0

\frac{1}{b^{2} + c^{2} - a^{2}} + \frac{1}{c^{2} + a^{2} - b^{2}} + \frac{1}{a^{2} + b^{2} - c^{2}}

\frac{a^{2} + b^{2} + c^{2}}{R^{2}}

8

a^{2} = b^{2} + c^{2}

b^{2} = c^{2} + a^{2}

c^{2} = a^{2} + b^{2}

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