Question

quadrilateral hijk was changed to create quadrilateral lmnp
determine if quadrilateral hijk is similar to quadrilateral lmnp.
a. quadrilateral hijk is similar to quadrilateral lmnp because dilation $(x,y)$ to $(0.25x,0.25y)$ maps quadrilateral hijk to quadrilateral lmnp.
b. quadrilateral hijk is not similar to quadrilateral lmnp because geometric stretch $(x,y)$ to $(0.5x,0.25y)$ maps quadrilateral hijk to quadrilateral lmnp.
c. quadrilateral hijk is not similar to quadrilateral lmnp because geometric stretch $(x,y)$ to $(0.25x,0.5y)$ maps quadrilateral hijk to quadrilateral lmnp.
d. quadrilateral hijk is similar to quadrilateral lmnp because dilation $(x,y)$ to $(0.5x,0.5y)$ maps quadrilateral hijk to quadrilateral lmnp.

Explanation:

Step1: Find the length of sides of HIJK

Assume \(H(-4,-2)\), \(I(-4,6)\), \(J(6,6)\), \(K(6,-2)\).
Length of \(HI = 6 - (-2)=8\), length of \(HK=6 - (-4) = 10\).

Step2: Find the length of sides of LMNP

Assume \(L(-2,-2)\), \(M(-2,2)\), \(N(2,2)\), \(P(2,-2)\).
Length of \(LM = 2 - (-2)=4\), length of \(LP=2 - (-2) = 4\).

Step3: Check the scale factor

For \(x\) - coordinate: \(\frac{-2-(-2)}{-4 - (-4)}=\frac{2 - (-2)}{6 - (-4)} = 0.5\) (using \(x\) - values of corresponding vertices).
For \(y\) - coordinate: \(\frac{2-(-2)}{6 - (-2)}=0.5\) (using \(y\) - values of corresponding vertices).
A dilation \((x,y)\to(0.5x,0.5y)\) maps \(HIJK\) to \(LMNP\). Similarity is preserved under dilation.

Answer:

D. Quadrilateral \(HIJK\) is similar to quadrilateral \(LMNP\) because dilation \((x,y)\) to \((0.5x,0.5y)\) maps quadrilateral \(HIJK\) to quadrilateral \(LMNP\).