Question

assessment (4.4 to 4.6): similarity and transformations

1. graph \\( \triangle a b c \\) with vertices \\( a ( - 2,1 ), b ( - 2, - 1 ) \\),

and \\( c ( 2, - 2 ) \\) and its image after a dilation with a
scale factor of 2.
(show work-1 point)

1. describe a similarity transformation that

maps \\( \triangle a b c \\) to \\( \triangle d e f \\).
a. reflection in the \\( y \\)-axis, followed by a dilation with a
scale factor of \\( \frac { 1 } { 4 } \\).
b. reflection in the \\( y \\)-axis, followed by a dilation with a
scale factor of 4.
c. reflection in the \\( x \\)-axis, followed by a dilation with a
scale factor of \\( \frac { 1 } { 4 } \\).
d. reflection in the \\( x \\)-axis, followed by a dilation with a
scale factor of 4.

1. write a coordinate rule for the dilation.

rule:

1. given \\( \triangle r s t \\) and \\( \triangle x y z \\). which sequence of

transformations shows \\( \triangle r s t \sim \triangle x y z \\)?
a. rotation of \\( 270 ^ { \circ } \\) clockwise about the origin followed
by a dilation centered at \\( r \\) with scale factor
of \\( \frac { 1 } { 3 } \\).
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 question 2

• First, check the orientation of \(\triangle ABC\) and \(\triangle DEF\). The \(y\) - axis reflection changes the \(x\) - coordinate sign (\((x,y)\to(-x,y)\)). But looking at the position of the triangles, we need to check the vertical orientation. A reflection in the \(x\) - axis changes the \(y\) - coordinate sign (\((x,y)\to(x, - y)\)).
• Then, calculate the scale factor. Let's assume a point \(A(x_1,y_1)\) in \(\triangle ABC\) and its corresponding point \(D(x_2,y_2)\) in \(\triangle DEF\). If we consider the distance from the \(x\) - axis (or use side - length ratios). Suppose the height of \(\triangle ABC\) is \(h_1\) and the height of \(\triangle DEF\) is \(h_2\). If \(h_1 = 1\) (say from \(y = 0\) to \(y=1\) for a vertex of \(\triangle ABC\)) and \(h_2 = 4\) (from \(y = 0\) to \(y = 4\) for the corresponding vertex of \(\triangle DEF\) after reflection), the scale factor \(k=\frac{h_2}{h_1}=4\). A reflection in the \(x\) - axis (\((x,y)\to(x, - y)\)) followed by a dilation with a scale factor of \(4\) (\((x,y)\to(4x,4y)\)) maps \(\triangle ABC\) to \(\triangle DEF\).

Step2: Analyze question 4

• For a rotation of \(270^{\circ}\) clockwise about the origin, the rotation rule is \((x,y)\to(y, - x)\).
• Then, consider the size change. If we assume a side - length of \(\triangle RST\) is \(l_1\) and a side - length of \(\triangle XYZ\) is \(l_2\). If \(l_2 = 3l_1\), a dilation with a scale factor of \(3\) centered at the origin (\((x,y)\to(3x,3y)\)) is needed after the rotation.

Answer:

1. D. reflection in the \(x\) - axis, followed by a dilation with a scale factor of \(4\).
2. B. Rotation of \(270^{\circ}\) clockwise about the origin followed by a dilation centered at \(R\) with scale factor of \(3\).