Question

analyzing compositions of dilations
the composition ( d_{o,0.75}(x,y)cdot d_{o,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 mcongangle 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 transformation that changes the size of a figure but not its shape. So, corresponding angles of the original triangle and the dilated triangle are congruent (\(\angle M\cong\angle M''\)), and the triangles are similar (\(\triangle LMN\sim\triangle L''M''N''\)).

Step2: Composition of Dilations

The composition of two dilations \(D_{O,0.75}(x,y)\cdot D_{O,2}(x,y)\) is equivalent to \(D_{O,0.75\times2}(x,y)=D_{O,1.5}(x,y)\).

• For vertex coordinates:
• Assume the original coordinates of \(L\) (from the graph, \(L(- 2,2)\)), \(M(-1,-1)\), \(N(2,-1)\)
• After dilation \(D_{O,1.5}(x,y)\):
• For \(L\): \(x=-2\times1.5=-3\), \(y = 2\times1.5=3\) (the statement about \(L''(-3,1.5)\) is wrong).
• For \(M\): \(x=-1\times1.5=-1.5\), \(y=-1\times1.5=-1.5\) (the statement about \(M''(1.5,-1.5)\) is wrong).
• For \(N\): \(x = 2\times1.5=3\), \(y=-1\times1.5=-1.5\) (the statement about \(N''(3,-1.5)\) is correct).
• Since dilation changes the size, \(\triangle LMN

ot\cong\triangle L''M''N''\)

Answer:

\(\angle M\cong\angle M''\), \(\triangle LMN\sim\triangle L''M''N''\), The coordinates of vertex \(N''\) are \((3,-1.5)\)