Question

question 10 of 10
how many degrees has △abc been rotated counterclockwise about the
origin?
graph of coordinate plane with △abc (blue) and △abc (red)
a. 360°
other options partially visible

Explanation:

Answer:

To determine the rotation angle, we analyze the coordinates of the original triangle \( \triangle ABC \) and its image \( \triangle A'B'C' \):

• Original points (approximate from the grid):
• \( A \): \( (3, -1) \) (or similar positive \( x \), negative \( y \))
• \( B \): \( (4, -6) \) (positive \( x \), negative \( y \))
• \( C \): \( (5, -1) \) (positive \( x \), negative \( y \))

• Image points (red triangle \( \triangle A'B'C' \)):
• \( A' \): \( (-2, 1) \) (negative \( x \), positive \( y \))
• \( B' \): \( (-4, 6) \) (negative \( x \), positive \( y \))
• \( C' \): \( (-6, 1) \) (negative \( x \), positive \( y \))

A counterclockwise rotation of \( 90^\circ \) transforms \( (x, y) \to (-y, x) \), but here the transformation is \( (x, -y) \to (-x, y) \), which is a \( 180^\circ \) rotation (since \( (x, y) \to (-x, -y) \) for \( 180^\circ \), but our original \( y \)-coordinates are negative, so \( (x, -y) \to (-x, y) \) is equivalent to \( 180^\circ \) rotation).

Wait, correcting: A \( 180^\circ \) counterclockwise (or clockwise) rotation about the origin transforms \( (x, y) \to (-x, -y) \). For our original triangle, if \( A \) is \( (3, -1) \), then \( -x = -3 \), \( -y = 1 \), so \( A' \) would be \( (-3, 1) \) (matching the grid’s \( A' \) near \( (-2, 1) \), likely due to grid precision). Similarly, \( B(4, -6) \to (-4, 6) \) (matches \( B' \) at \( (-4, 6) \)), and \( C(5, -1) \to (-5, 1) \) (matches \( C' \) near \( (-6, 1) \)).

Thus, the rotation is \( 180^\circ \). (Assuming the missing option is \( 180^\circ \); if options include \( 180^\circ \), that is the answer.)

(Note: If the visible options include \( 180^\circ \), the final answer is \( \boldsymbol{180^\circ} \).)