1. On the coordinate plane, \( \triangle X Y Z \) has vertices of \( X(0,4), Y(2,-1) \), and \( Z(-4,2) \). What are the vertex coordinates of the triangle after a dilation centered at the origin with a scale factor of 5 ? \( X^{\prime}(0,8), Y^{\prime}(20,-10), Z^{\prime}(-8,4) \) \( X^{\prime}(0,12), Y^{\prime}(6,-3), Z^{\prime}(-12,6) \) \( X^{\prime}(5,9), Y^{\prime}(7,4), Z^{\prime}(1,7) \) \( X^{\prime}(0,20), Y^{\prime}(10,-5), Z^{\prime}(-20,10) \)

To determine the new coordinates of the vertices after a dilation centered at the origin with a scale factor of 5, you simply multiply each coordinate by 5. For point \( X(0,4) \): - \( X^{\prime}(0 \cdot 5, 4 \cdot 5) = (0, 20) \). For point \( Y(2,-1) \): - \( Y^{\prime}(2 \cdot 5, -1 \cdot 5) = (10, -5) \). For point \( Z(-4,2) \): - \( Z^{\prime}(-4 \cdot 5, 2 \cdot 5) = (-20, 10) \). Thus, the new vertices are \( X^{\prime}(0, 20), Y^{\prime}(10, -5), Z^{\prime}(-20, 10) \). So, the answer is: \( X^{\prime}(0,20), Y^{\prime}(10,-5), Z^{\prime}(-20,10) \). --- Dilation is a fabulous concept in geometry! It helps scale shapes up or down while preserving their proportions. Think of it as blowing up a balloon; the shape remains the same, just larger or smaller! Real-world applications of dilation can be seen in photography and computer graphics, where images are resized but need to maintain clarity and proportions. So next time you adjust the size of a photo, remember you might just be performing a dilation!