Question

which rotation is shown in the coordinate plane? 180° clockwise 180° counterclockwise 90° counterclockwise 90° clockwise

Explanation:

Step1: Recall Rotation Rules

For a \(180^\circ\) rotation (clockwise or counterclockwise), the rule is \((x,y)\to(-x,-y)\). For a \(90^\circ\) counterclockwise rotation, the rule is \((x,y)\to(-y,x)\), and for a \(90^\circ\) clockwise rotation, the rule is \((x,y)\to(y,-x)\).

Step2: Analyze Coordinates (Assume Original and Image Points)

Let's identify points. Let's say original triangle has points \(D\), \(E\), \(F\), and image has \(D'\), \(E'\), \(F'\).
- For a \(180^\circ\) rotation, each point \((x,y)\) should map to \((-x,-y)\). Let's check a point, say \(E\) (assuming \(E\) is at \((4, -1)\) (looking at grid), then \(E'\) should be at \((-4, 1)\)? Wait, no, looking at the grid, \(E\) is at \((4, -1)\)? Wait, no, the y-axis: below x-axis is negative. Wait, \(E\) is at \((4, -1)\)? No, wait, \(E\) is on the x-axis? Wait, no, \(E\) is at \((4, -1)\)? Wait, no, the image \(E'\) is at \((-4, -1)\)? Wait, no, let's look again. Wait, original triangle: \(D\) is at \((-1, -2)\)? Wait, no, the grid: \(D\) is at \((-1, -2)\)? Wait, no, the right triangle: \(D\) is at \((-1, -2)\)? No, the right triangle (original) has \(D\) at \((-1, -2)\)? Wait, no, the left triangle (image) has \(D'\) at \((-3, -1)\)? Wait, maybe better to use the \(180^\circ\) rotation property: a \(180^\circ\) rotation turns the figure so that it is upside - down and reversed, and the direction (clockwise or counterclockwise for \(180^\circ\)) doesn't matter as \(180^\circ\) clockwise and counterclockwise rotations are the same. Wait, no, actually \(180^\circ\) clockwise and counterclockwise rotations have the same result. But let's check the other options. A \(90^\circ\) rotation would change the orientation (vertical/horizontal sides), but here the triangle's orientation (e.g., sides: original has a vertical side \(EF\), image has a horizontal side \(E'F'\)? Wait, no, original \(EF\) is vertical (from \(E(4, -1)\) to \(F(4, -3)\) maybe), and image \(E'F'\) is horizontal (from \(E'(-4, -1)\) to \(F'(-2, -3)\)? Wait, no, maybe better to see that a \(180^\circ\) rotation maps each point to the opposite side of the origin. Let's take point \(D\): suppose \(D\) is at \((2, -2)\), then \(D'\) should be at \((-2, 2)\)? No, wait, looking at the grid, the original triangle (right) and image (left) are related by \(180^\circ\) rotation. Because a \(90^\circ\) rotation would change the side from vertical to horizontal or vice versa, but here the triangle's shape (a triangle with a vertical side becomes a triangle with a horizontal side? No, wait, no: the original triangle (right) has a vertical side \(EF\) (from \(E(4, -1)\) to \(F(4, -3)\)), and the image triangle (left) has a horizontal side \(E'F'\) (from \(E'(-4, -1)\) to \(F'(-2, -3)\))? No, that's not. Wait, no, actually, a \(180^\circ\) rotation: if you rotate a figure \(180^\circ\) around the origin, the figure is rotated halfway around the circle, so each point \((x,y)\) goes to \((-x,-y)\). Let's check the center: the origin? Wait, the triangles are symmetric with respect to the origin? Let's take point \(E\): if \(E\) is at \((4, -1)\), then \(E'\) should be at \((-4, 1)\)? But in the image, \(E'\) is at \((-4, -1)\)? Wait, no, maybe I got the y - coordinate wrong. Wait, the x - axis: right is positive, left is negative. Y - axis: up is positive, down is negative. So \(E\) is at \((4, -1)\) (down from x - axis), and \(E'\) is at \((-4, -1)\)? No, that's a reflection? No, wait, no, the image triangle is on the left, and original on the right. Wait, maybe the rotation is \(180^\circ\) clockwise (or counterclockwise, since \(180^\circ\) is same for both). Wait, the other options: \(90^\circ\) counterclockwise or clockwise would change the orientation (e.g., a vertical side would become horizontal). But here, the triangle's orientation (the angle) is reversed by \(180^\circ\), not \(90^\circ\). So the rotation is \(180^\circ\) clockwise (or counterclockwise, but the option is \(180^\circ\) clockwise? Wait, no, \(180^\circ\) clockwise and counterclockwise are the same. But looking at the options, the first option is \(180^\circ\) clockwise. Wait, let's confirm: a \(180^\circ\) rotation maps each point to the opposite side of the origin, so the figure is rotated halfway, which matches the image here. The \(90^\circ\) rotations would change the direction (e.g., a right - angled triangle with vertical leg would have horizontal leg after \(90^\circ\) rotation), but here the legs' directions are reversed by \(180^\circ\), not \(90^\circ\). So the correct rotation is \(180^\circ\) clockwise.

Answer:

180° clockwise