Question

figure efghij was changed to create figure klmnpq. determine if figure efghij is similar to figure klmnpq. a. figure efghij is not similar to figure klmnpq because geometric stretch (x,y) to (2x,1.5y) maps figure efghij to figure klmnpq. b. figure efghij is not similar to figure klmnpq because geometric stretch (x,y) to (1.5x,2y) maps figure efghij to figure klmnpq. c. figure efghij is similar to figure klmnpq because dilation (x,y) to (1.5x,1.5y) maps figure efghij to figure klmnpq. d. figure efghij is similar to figure klmnpq because dilation (x,y) to (2x,2y) maps figure efghij to figure klmnpq.

Explanation:

Step1: Recall the definition of similar figures

Similar figures have all corresponding angles equal and all corresponding sides in proportion (a dilation with a scale factor \(k\) for both \(x\) and \(y\) - coordinates). A geometric stretch (\(x,y\) to \((ax,by)\) where \(a
eq b\)) changes the shape (ratios of side - lengths) and does not preserve similarity. A dilation (\(x,y\) to \((kx,ky)\) where \(k>0\)) preserves similarity.

Step2: Analyze each option

• Option A: A geometric stretch \((x,y)\to(2x,1.5y)\) (\(a = 2\), \(b=1.5\), \(a

eq b\)) changes the shape (ratios of side - lengths). So, the figures are not similar.

• Option B: A geometric stretch \((x,y)\to(1.5x,2y)\) (\(a = 1.5\), \(b = 2\), \(a

eq b\)) changes the shape (ratios of side - lengths). So, the figures are not similar.

• Option C: Assume a point \(E(-4,2)\) in \(EFGHIJ\). If we apply \((x,y)\to(1.5x,1.5y)\), the image of \(E\) is \((-4\times1.5,2\times1.5)=(-6,3)\). But in the second figure, the corresponding point \(K\) is not at \((-6,3)\).
• Option D: Assume a point \(E(-4,2)\) in \(EFGHIJ\). If we apply \((x,y)\to(2x,2y)\), the image of \(E\) is \((-4\times2,2\times2)=(-8,4)\) (incorrect). Wait, let's take another approach. Count the side - lengths.

Let's consider the horizontal and vertical components of the sides. For a dilation \((x,y)\to(2x,2y)\), all side - lengths of \(EFGHIJ\) are multiplied by \(2\). The angles remain the same (since dilation is a similarity transformation that preserves angle measures). A dilation \((x,y)\to(2x,2y)\) (a scale factor \(k = 2\) for both \(x\) and \(y\)) is a similarity transformation.

Answer:

D. Figure \(EFGHIJ\) is similar to figure \(KLMNPQ\) because dilation \((x,y)\) to \((2x,2y)\) maps figure \(EFGHIJ\) to figure \(KLMNPQ\)