Question

4.6): similarity and transformations

1. write a coordinate rule for the dilation.

rule:

1. given ( \triangle rst ) and ( \triangle xyz ). which sequence of

a. rotation of ( 270^{circ} ) clockwise about the origin followed
by a dilation centered at ( r ) with scale factor
of ( \frac{1}{2} ).
b. rotation of ( 270^{circ} ) clockwise about the origin
followed by a dilation centered at ( r ) with scale factor
of 3.
c. dilation centered at the origin with scale factor
of 3 followed by a rotation of ( 270^{circ} ) counterclockwise
about point ( t ).
d. dilation centered at the origin with scale factor
of ( \frac{1}{3} ) followed by a rotation of ( 270^{circ} ) counterclockwise
about point ( t ).

Explanation:

Step1: Analyze the size relationship

First, observe the size of \(\triangle RST\) and \(\triangle XYZ\). We can see that \(\triangle XYZ\) is larger than \(\triangle RST\). If we assume the side - length of \(\triangle RST\) is \(l_1\) and the side - length of \(\triangle XYZ\) is \(l_2\), from the graph (by counting the grid units or using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for corresponding vertices), we find that the scale factor \(k\) of the dilation (if centered at the origin) should satisfy \(l_2 = 3l_1\).

Step2: Analyze the rotation direction

For a rotation of \(270^{\circ}\) clockwise about the origin, the rotation rule for a point \((x,y)\) is \((x,y)\to(y, - x)\). For a rotation of \(270^{\circ}\) counter - clockwise about a point (not the origin), it is more complex. But if we consider the general transformation of similarity. A dilation centered at the origin with a scale factor of \(3\) will enlarge the figure. And a \(270^{\circ}\) clockwise rotation about the origin is a standard rotation.

Let's check each option:

• Option A: A scale factor of \(\frac{1}{2}\) will shrink the figure, so A is wrong.
• Option B: A rotation of \(270^{\circ}\) clockwise about the origin (a standard rotation) followed by a dilation centered at \(R\) (non - origin center for dilation is more complex, but if we consider the overall similarity transformation, and the scale factor of \(3\) which is correct for the size relationship).
• Option C: A dilation followed by a rotation about point \(T\) (non - origin rotation center for the second transformation makes the transformation non - standard for similarity in the way described in the problem's context of simple similarity transformations shown in basic geometry problems).
• Option D: A scale factor of \(\frac{1}{3}\) will shrink the figure, so D is wrong.

Answer:

B. Rotation of \(270^{\circ}\) clockwise about the origin followed by a dilation centered at \(R\) with scale factor of \(3\).