Question

question 2 of 10

what degree of rotation about the origin will cause the triangle below to map onto itself?

a. \\(360^\circ\\)
b. \\(270^\circ\\)
c. \\(90^\circ\\)
d. \\(180^\circ\\)

Explanation:

Identify the coordinates of the vertices

The vertices of triangle \(ABC\) are:

$$ A(-8, 1), \quad B(-5, 4), \quad C(-3, 1) $$

Analyze rotational symmetry about the origin

A non-trivial rotation (such as \(90^\circ\), \(180^\circ\), or \(270^\circ\)) about the origin \((0,0)\) maps the vertices to different quadrants:

• \(90^\circ\) counterclockwise: \((x, y) \to (-y, x)\)
• \(180^\circ\): \((x, y) \to (-x, -y)\)
• \(270^\circ\) counterclockwise: \((x, y) \to (y, -x)\)

Since the triangle is not centered at the origin and does not have rotational symmetry about the origin, only a full rotation will map it onto itself.

Determine the required angle of rotation

A full rotation of \(360^\circ\) about any point, including the origin, maps every geometric figure exactly onto itself:

$$ (x, y) \to (x, y) $$

Answer:

• A. \(360^\circ\) (Correct answer)
• B. \(270^\circ\)
• C. \(90^\circ\)
• D. \(180^\circ\)