This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for problems 9 and 10. Problem 9: Graph ABC with vertices A(-4, 4), B(-1, 7), and C(-1, 4) and its image after a 270° clockwise rotation about the origin. Step 1: Identify the original vertices. The vertices of ABC are A(-4, 4), B(-1, 7), and C(-1, 4). Step 2: Determine the transformation rule. A 270° clockwise rotation about the origin (0,0) transforms a point (x, y) to (y, -x). Step 3: Apply the transformation rule to find the image vertices. For vertex A(-4, 4): A' = (4, -(-4)) = (4, 4) For vertex B(-1, 7): B' = (7, -(-1)) = (7, 1) For vertex C(-1, 4): C' = (4, -(-1)) = (4, 1) The vertices of the image A'B'C' are A'(4, 4), B'(7, 1), and C'(4, 1). Step 4: Graph the triangles. To graph, plot the original vertices A(-4, 4), B(-1, 7), C(-1, 4) and connect them to form ABC. Then, plot the image vertices A'(4, 4), B'(7, 1), C'(4, 1) and connect them to form A'B'C'. The coordinates of the image are: A'(4, 4), B'(7, 1), C'(4, 1) --- Problem 10: A rectangle has vertices at A(-3, 1), B(-1, 1), C(-1, 4), and D(-3, 4). Reflect the rectangle across the line y = 2 and find the coordinates of the new vertices. Explain how the position of the shape changes after the reflection. Step 1: Identify the original vertices. The vertices of the rectangle are A(-3, 1), B(-1, 1), C(-1, 4), and D(-3, 4). Step 2: Determine the transformation rule for reflection across y=2. A reflection across the horizontal line y=k transforms a point (x, y) to (x, 2k - y). Here, k=2, so the rule is (x, y) (x, 2(2) - y) = (x, 4 - y). Step 3: Apply the transformation rule to find the new vertices. For vertex A(-3, 1): A' = (-3, 4 - 1) = (-3, 3) For vertex B(-1, 1): B' = (-1, 4 - 1) = (-1, 3) For vertex C(-1, 4): C' = (-1, 4 - 4) = (-1, 0) For vertex D(-3, 4): D' = (-3, 4 - 4) = (-3, 0) The coordinates of the new vertices are: A'(-3, 3), B'(-1, 3), C'(-1, 0), D'(-3, 0) Step 4: Explain how the position of the shape changes after the reflection. The reflection across the line y=2 flips the rectangle vertically. Points that were below the line y=2 are now above it, and points that were above the line y=2 are now below it. The x-coordinates of the vertices remain unchanged, while the y-coordinates are transformed such that their distance from the line y=2 is preserved but on the opposite side. Step 5: Explain how the shape changes after the reflection. Reflection is an isometry, which means it preserves the size and shape of the figure. The rectangle remains a rectangle of the same dimensions (length and width) and its angles remain 90 degrees. Only its position and orientation in the coordinate plane change. Last free one today — make it count tomorrow, or type /upgrade for unlimited.