Question

1. \\( \triangle dog \\) and \\( \triangle bat \\) are shown on the coordinate plane. select all statements that explain why \\( \triangle dog \\) and \\( \triangle bat \\) are similar.

a \\( \triangle dog \\) can be mapped onto \\( \triangle bat \\) by a \\( 180^{circ} \\) rotation about the origin, followed by a dilation with a scale factor of \\( \frac{1}{3} \\) with the center at the origin.
b \\( \triangle dog \\) can be mapped onto \\( \triangle bat \\) by a reflection over the \\( x \\)-axis, followed by a dilation with a scale factor of \\( \frac{1}{3} \\) with the center at the origin.
c \\( \triangle dog \\) can be mapped onto \\( \triangle bat \\) by a dilation of \\( \frac{1}{3} \\) with the center at the origin, followed by a reflection over the \\( y \\)-axis.
d \\( \triangle bat \\) can be mapped onto \\( \triangle dog \\) by a reflection over they \\( y \\)-axis, followed by a dilation with a scale factor of 3 with the center at the origin.
e \\( \triangle bat \\) can be mapped onto \\( \triangle dog \\) by a dilation with a scale factor of 3 with the center at the origin, followed by a reflection over the \\( x \\)-axis.

Explanation:

Step1: Analyze Option A

A \(180^{\circ}\) rotation about the origin changes the sign of both \(x\) and \(y\) coordinates. Then a dilation with scale factor \(\frac{1}{3}\) would reduce the size. But visually and by coordinate - mapping (assuming we could assign coordinates), this is not the correct transformation.

Step2: Analyze Option B

A reflection over the \(x\) - axis changes the sign of the \(y\) - coordinate. Then a dilation with scale factor \(\frac{1}{3}\) would reduce the size. If we assume coordinates (say \(D(x_1,y_1)\), \(O(x_2,y_2)\), \(G(x_3,y_3)\) and \(B(x_1',y_1')\), \(A(x_2',y_2')\), \(T(x_3',y_3')\)), reflecting \(\triangle DOG\) over the \(x\) - axis and then dilating by \(\frac{1}{3}\) can map it to \(\triangle BAT\).

Step3: Analyze Option C

A dilation of \(\frac{1}{3}\) first and then a reflection over the \(y\) - axis (changes the sign of the \(x\) - coordinate) is not the correct sequence of transformations to map \(\triangle DOG\) to \(\triangle BAT\).

Step4: Analyze Option D

A reflection over the \(y\) - axis (changes the sign of the \(x\) - coordinate) and then a dilation with scale factor \(3\). If we reverse the mapping (mapping \(\triangle BAT\) to \(\triangle DOG\)), reflecting \(\triangle BAT\) over the \(y\) - axis and then dilating by \(3\) can map it to \(\triangle DOG\).

Step5: Analyze Option E

A dilation with scale factor \(3\) first and then a reflection over the \(x\) - axis (changes the sign of the \(y\) - coordinate) is the correct sequence to map \(\triangle BAT\) to \(\triangle DOG\).

Answer:

B. \(\triangle DOG\) can be mapped onto \(\triangle BAT\) by a reflection over the \(x\) - axis, followed by a dilation with a scale factor of \(\frac{1}{3}\) with the center at the origin.
D. \(\triangle BAT\) can be mapped onto \(\triangle DOG\) by a reflection over the \(y\) - axis, followed by a dilation with a scale factor of \(3\) with the center at the origin.
E. \(\triangle BAT\) can be mapped onto \(\triangle DOG\) by a dilation with a scale factor of \(3\) with the center at the origin, followed by a reflection over the \(x\) - axis.