Question

Using the rule Ro, 180°, what are the vertices of Triangle QRS after being rotated?

A) Q(1,0) → R(5,0) → S(3,5)
B) Q(1,0) → R(5,0) → S(3,-5)
C) Q(1,0) → R(-5,0) → S(-3,5)
D) Q(1,0) → R(-5,0) → S(-3,-5)

Answer 1

Using the rule Ro, 180°, each vertex of the triangle is rotated 180 degrees, resulting in new coordinates which are the negation of the original coordinates. The vertices of Triangle QRS after being rotated are Q'(-1,0), R'(-5,0), and S'(-3,-5), which corresponds to option D.

Step-by-step explanation:

The rule Ro, 180° indicates a rotation of 180 degrees around the origin. To find the coordinates of the vertices after the rotation, we can apply the transformation equations:

• To transform x, the formula is x' = x cos θ + y sin θ. For a 180-degree rotation, θ equals 180 degrees, and the cos(180°) = -1, sin(180°) = 0. Hence, x' = -x.
• To transform y, the formula is y' = -x sin φ + y cos φ. Similarly, for φ equaling 180 degrees, sin(180°) = 0 and cos(180°) = -1, leading to y' = -y.

Applying these transformations to Triangle QRS:

• Point Q(1,0) becomes Q'(-1,0)
• Point R(5,0) becomes R'(-5,0)
• Point S(3,5) becomes S'(-3,-5)

Therefore, the correct vertices after the rotation are D) Q'(-1,0) → R'(-5,0) → S'(-3,-5).