Question What is the image point of \( (1,7) \) after the transformation \( R_{180^{\circ}} \circ D_{2} \) ?

To find the image point of \( (1,7) \) after the transformation \( R_{180^{\circ}} \circ D_{2} \), we will apply the transformations step by step. First, we reflect the point \( (1,7) \) over the line \( x=2 \). The reflection (or direct distance) means that if the original point is 1 unit to the left of the line \( x=2 \), it will be 1 unit to the right after reflection. This yields the new point \( (3, 7) \). Next, we perform a rotation of \( 180^{\circ} \) around the origin. The formula for a \( 180^{\circ} \) rotation is \( (x, y) \rightarrow (-x, -y) \). Applying this to our point \( (3, 7) \) gives us the image point \( (-3, -7) \). To summarize, the image point of \( (1,7) \) after the transformation \( R_{180^{\circ}} \circ D_{2} \) is \( (-3, -7) \). Now, here are two fun facts related to transformations: Reflective symmetry is a concept that appears everywhere in nature and art! Think about butterflies, which are often perfect reflections of each other. This beauty isn't just for aesthetics; understanding symmetry helps architects create buildings that not only look great but also withstand structural stress better. Ever tried to transform a point but stumbled upon the calculations? It's a common pitfall! Always double-check the sequence of transformations you're applying. Getting them in the wrong order can completely change your final point! A good rule of thumb: think about the transformations as layers in a cake – they all stack but need to be in the right order for that perfect slice!