Question

analyzing compositions of dilations
the composition ( d_{0,0.75}(x,y)cdot d_{0,2}(x,y) ) is applied to ( \triangle lmn ) to create ( \triangle lmn ).
which statements must be true regarding the two triangles? check all that apply.
( angle m=angle m )
( \triangle lmnsim\triangle lmn )
( \triangle lmncong\triangle lmn )
the coordinates of vertex ( l ) are ( (-3,1.5) ).
the coordinates of vertex ( n ) are ( (3,-1.5) ).
the coordinates of vertex ( m ) are ( (1.5,-1.5) ).

Explanation:

Step1: Properties of dilation

Dilation is a similarity transformation. When a figure is dilated, the corresponding angles are congruent. So, \(\angle M=\angle M''\). Also, if two figures are the result of a series of dilations (composite dilation), the original figure and the final figure are similar. So, \(\triangle LMN\sim\triangle L''M''N''\).

Step2: Composite - dilation formula

The formula for a composite dilation \(D_{O,k_1}\circ D_{O,k_2}(x,y)=D_{O,k_1\times k_2}(x,y)\). Here \(k_1 = 0.75\) and \(k_2=0.2\), so \(k=0.75\times0.2 = 0.15\).
If \(L(- 2,2)\), then \(L''=D_{O,0.15}(-2,2)=(-2\times0.15,2\times0.15)=(-0.3,0.3)\).
If \(N(2,-1)\), then \(N''=D_{O,0.15}(2,-1)=(2\times0.15,-1\times0.15)=(0.3,-0.15)\).
If \(M(-1,-1)\), then \(M''=D_{O,0.15}(-1,-1)=(-1\times0.15,-1\times0.15)=(-0.15,-0.15)\).

Answer:

\(\angle M=\angle M''\), \(\triangle LMN\sim\triangle L''M''N''\)