Question

question 1 - 11
select all the transformations that do not preserve distance but preserve angle measure
reflection across the line ( y = x )
translation of 3 units upwards
counterclockwise rotation of ( 180^{circ} ) about the origin
dilation by a scale factor of 0.5 about the origin
dilation by a scale factor of - 1 about the point ( (1,1) )
translation of 2 units to the right then dilation by a scale factor of 1.5 about the point ( (1,1) )

Explanation:

Step1: Recall properties of transformations

• Reflection: Reflection across a line (e.g., \(y = x\)) is a rigid transformation. Rigid transformations preserve both distance and angle measure.
• Translation: Translation (e.g., 3 units upwards or 2 units to the right) is a rigid transformation. Rigid transformations preserve both distance and angle measure.
• Rotation: Rotation (e.g., counter - clockwise rotation of \(180^{\circ}\) about the origin) is a rigid transformation. Rigid transformations preserve both distance and angle measure.
• Dilation: Dilation by a scale factor \(k

eq1\) (e.g., dilation by a scale factor of \(0.5\) about the origin, dilation by a scale factor of \(1.5\) about the point \((1,1)\)) is a non - rigid transformation. A dilation with scale factor \(k\) changes the distance between points (\(d'=|k|d\), where \(d\) is the original distance and \(d'\) is the new distance) but preserves angle measure. A dilation with scale factor \(k = - 1\) (e.g., dilation by a scale factor of \(-1\) about the point \((1,1)\)) is equivalent to a rotation of \(180^{\circ}\) about the point \((1,1)\) (since \(x'=(1)+(-1)(x - 1)=2 - x\) and \(y'=(1)+(-1)(y - 1)=2 - y\)), which is a rigid transformation (preserves distance and angle measure).

Step2: Analyze each option

• For the dilation by a scale factor of \(0.5\) about the origin:

Let \(P(x,y)\) be a point. After dilation \(P'(0.5x,0.5y)\). The distance from the origin \(d=\sqrt{x^{2}+y^{2}}\) and \(d'=\sqrt{(0.5x)^{2}+(0.5y)^{2}}=0.5\sqrt{x^{2}+y^{2}}\). But for two lines intersecting at an angle \(\theta\), the angle between their images after dilation is the same as \(\theta\).

• For the translation of 2 units to the right then dilation by a scale factor of \(1.5\) about the point \((1,1)\):

Let \(P(x,y)\) be a point. After translation \(P_1(x + 2,y)\). Then after dilation \(P_2=(1)+1.5((x + 2)-1,(y)-1)=(1.5x+2.5,1.5y - 0.5)\). The distance between two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) changes (e.g., if \(P=(0,0)\) and \(Q=(2,0)\), after translation \(P_1=(2,0)\), \(Q_1=(4,0)\), after dilation \(P_2=(2.5,- 0.5)\), \(Q_2=(4.5,-0.5)\). The original distance \(d=\sqrt{(2 - 0)^{2}+(0 - 0)^{2}}=2\), the new distance \(d'=\sqrt{(4.5 - 2.5)^{2}+(-0.5+0.5)^{2}}=2\) (this is a coincidence for this simple case, in general \(d'=|k|d\) for dilation part). But for angles, since dilation is a similarity transformation (preserves angle measure) and translation is a rigid transformation (preserves angle measure).

Answer:

Dilation by a scale factor of \(0.5\) about the origin, Translation of 2 units to the right then dilation by a scale factor of \(1.5\) about the point \((1,1)\)