Question

problem 4 pg. 60 a) reflect trapezoid a over the x - axis. label the image trapezoid b. reflect trapezoid b over the y - axis. label the image c. b) are trapezoids a and c congruent? explain how you know

Explanation:

Step1: Recall reflection rules

Reflection over x - axis: $(x,y)\to(x, - y)$. Reflection over y - axis: $(x,y)\to(-x,y)$.

Step2: Reflect trapezoid A over x - axis

Let the vertices of trapezoid A be $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. After reflection over x - axis, the new vertices of trapezoid B are $(x_1,-y_1),(x_2,-y_2),(x_3,-y_3),(x_4,-y_4)$. Plot and label trapezoid B.

Step3: Reflect trapezoid B over y - axis

Take the vertices of trapezoid B $(x_1,-y_1),(x_2,-y_2),(x_3,-y_3),(x_4,-y_4)$. After reflection over y - axis, the new vertices of trapezoid C are $(-x_1,-y_1),(-x_2,-y_2),(-x_3,-y_3),(-x_4,-y_4)$. Plot and label trapezoid C.

Step4: Check congruence

Two figures are congruent if they have the same shape and size. Since reflections are rigid motions (isometries) that preserve distance and angle measures, trapezoid A and trapezoid C are congruent because a sequence of reflections (reflection over x - axis and then over y - axis) is a rigid motion.

Answer:

a) Follow the reflection rules to plot trapezoid B (reflection of A over x - axis) and trapezoid C (reflection of B over y - axis).
b) Trapezoids A and C are congruent because reflections are rigid motions that preserve shape and size.