Question

Quadrilateral DEFG has vertices D(−2,4) , E(4,7) , F(10,3) , and G(8,0) .

What are the coordinates of D′E′F′G′ after quadrilateral DEFG has undergone a rotation 270° counterclockwise about the origin?

Enter your answers, as ordered pairs (x, y) , in the boxes.

Answer 1

To rotate a point counterclockwise about the origin, we can use the formulas x' = x*cos(theta) - y*sin(theta) and y' = x*sin(theta) + y*cos(theta). Applying these formulas, the new coordinates of D', E', F', and G' are (4, 2), (7, -4), (3, -10), and (0, -8), respectively.

Step-by-step explanation:

To rotate a point counterclockwise about the origin, we can use the following formulas:

x' = x*cos(theta) - y*sin(theta)

y' = x*sin(theta) + y*cos(theta)

Using these formulas, we can find the new coordinates of each vertex:

• D': x' = (-2)*cos(270°) - 4*sin(270°) = 4, y' = (-2)*sin(270°) + 4*cos(270°) = 2
• E': x' = 4*cos(270°) - 7*sin(270°) = 7, y' = 4*sin(270°) + 7*cos(270°) = -4
• F': x' = 10*cos(270°) - 3*sin(270°) = 3, y' = 10*sin(270°) + 3*cos(270°) = -10
• G': x' = 8*cos(270°) - 0*sin(270°) = 0, y' = 8*sin(270°) + 0*cos(270°) = -8

Therefore, the coordinates of D', E', F', G' are (4, 2), (7, -4), (3, -10), and (0, -8), respectively.

Answer 2

A rotation 270° counterclockwise about the origin is the same as rotation 90° clockwise about the origin and has a rule:

(x,y)→(y,-x).

Then:

• D(−2,4)→D'(4,2)
• E(4,7)→E'(7,-4)
• F(10,3)→F'(3,-10)
• G(8,0)→G'(0,-8)

Answer: the coordinates of vertices of quadrilateral D′E′F′G′ are D'(4,2), E'(7,-4), F'(3,-10), G'(0,-8).