Estimate parameter [color=#1e84cc][i]a[/i][/color] so that the matrix [color=#0000ff][i]M[/i][/color] represents reflection in line. [br][center][/center][center][img]data:image/png;base64,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[/img][/center]

1. method: Rotation is a direct isometry, hence |A| = 1, i.e. [math]a^2+\frac{1}{4}=1\Longrightarrow a=\pm\frac{\sqrt{3}}{2}[/math].[br]2. method (experimental): Use tool slider[icon]/images/ggb/toolbar/mode_slider.png[/icon]for unknown parameter [color=#1e84cc][i]a[/i][/color]. Define one parameter family of matrices M([color=#1e84cc][i]a[/i][/color]). [br][code]M ={{-0.5,a},{a,0.5}}[br][br][/code][br]Draw arbitrary object [color=#1e84cc][i]B[/i][/color] (point, segment or picture) and its image [color=#1e84cc][i]B[/i]'[/color] - GeoGebra command [code]ApplyMatrix(matrix,object)[/code]. Observe the effect of changing the slider [color=#1e84cc][i]a[/i][/color] and estimate correct value for parameter [color=#1e84cc][i]a[/i][/color]. [br][br]Experimental method is efficient for determination of fixed point and directions. Compare the position of arbitrary movable point [i][color=#1e84cc]B[/color][/i] and its image [color=#1e84cc][i]B'[/i][/color]. Find out the location where points coincide, [i][color=#1e84cc] B[/color][/i] = [color=#1e84cc][i]B[/i]'[/color]. There is the fixed point of transformation. The same method applyed on line [color=#1e84cc][i]f[/i][/color] gives you fixed direction. You should find the position where [i][color=#1e84cc]f[/color][/i] is parallel with image [color=#1e84cc][i]f'[/i][/color].[br]