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.
ScanSolve AI Answer
Here are the steps for the composite transformation: Step 1: Reflect quadrilateral PQRS in the x-axis to obtain P'Q'R'S'. The coordinate rule for reflection in the x-axis is (x, y) (x, -y). Given vertices P(-2, 1), Q(0, 3), R(2, 2), and S(1, 0): P' = (-2, -1) Q' = (0, -3) R' = (2, -2) S' = (1, 0) The coordinates of P'Q'R'S' are P'(-2, -1), Q'(0, -3), R'(2, -2), and S'(1, 0). Step 2: Translate P'Q'R'S' using the vector to obtain P''Q''R''S''. The translation vector is missing from the problem statement. For demonstration purposes, I will assume a translation vector of 1 \\ 1 . The coordinate rule for this translation is (x, y) (x+1, y+1). Applying the rule to the vertices P'(-2, -1), Q'(0, -3), R'(2, -2), and S'(1, 0): P'' = (-2+1, -1+1) = (-1, 0) Q'' = (0+1, -3+1) = (1, -2) R'' = (2+1, -2+1) = (3, -1) S'' = (1+1, 0+1) = (2, 1) The coordinates of P''Q''R''S'' are P''(-1, 0), Q''(1, -2), R''(3, -1), and S''(2, 1). Step 3: Rotate P''Q''R''S'' through 180^ about the origin to obtain P'''Q'''R'''S'''. The coordinate rule for a 180^ rotation about the origin is (x, y) (-x, -y). Applying the rule to the vertices P''(-1, 0), Q''(1, -2), R''(3, -1), and S''(2, 1): P''' = (-(-1), -(0)) = (1, 0) Q''' = (-(1), -(-2)) = (-1, 2) R''' = (-(3), -(-1)) = (-3, 1) S''' = (-(2), -(1)) = (-2, -1) The coordinates of P'''Q'''R'''S''' are P'''(1, 0), Q'''(-1, 2), R'''(-3, 1), and S'''(-2, -1). Step 4: Enlarge P'''Q'''R'''S''' with a scale factor of k = (1)/(2) centred at the origin to obtain P''''Q''''R''''S''''. The coordinate rule for enlargement with scale factor k=(1)/(2) centred at the origin is (x, y) ((1)/(2)x, (1)/(2)y). Applying the rule to the vertices P'''(1, 0), Q'''(-1, 2), R'''(-3, 1), and S'''(-2, -1): P'''' = ((1)/(2) × 1, (1)/(2) × 0) = ((1)/(2), 0) Q'''' = ((1)/(2) × -1, (1)/(2) × 2) = (-(1)/(2), 1) R'''' = ((1)/(2) × -3, (1)/(2) × 1) = (-(3)/(2), (1)/(2)) S'''' = ((1)/(2) × -2, (1)/(2) × -1) = (-1, -(1)/(2)) The coordinates of P''''Q''''R''''S'''' are P''''((1)/(2), 0), Q''''(-(1)/(2), 1), R''''(-(3)/(2), (1)/(2)), and S''''(-1, -(1)/(2)). To create the labeled graph, plot the original quadrilateral PQRS and all the transformed images P'Q'R'S', P''Q''R''S'', P'''Q'''R'''S''', and P''''Q''''R''''S'''' using the coordinates calculated above on a coordinate plane. Step 5: Compare the area relationship between PQRS and P''''Q''''R''''S''''. Reflection, translation, and rotation are rigid transformations, meaning they preserve the area of the shape (the area scale factor is 1). Enlargement with a scale factor k changes the area by a factor of k^2. In this sequence of transformations, the only transformation that changes the area is the final enlargement with a scale factor of k = (1)/(2). Therefore, the area of the final image P''''Q''''R''''S'''' is k^2 times the area of the original quadrilateral PQRS. Area(P''''Q''''R''''S'''') = ((1)/(2))^2 × Area(PQRS) Area(P''''Q''''R''''S'''') = (1)/(4) × Area(PQRS) The area of P''''Q''''R''''S'''' is one-fourth the area of PQRS. 3 done, 2 left today. You're making progress.