Question

maria drew the triangle that is shown.
maria rotated the triangle 90 degrees clockwise about the origin. what is the new triangle?

Explanation:

Step1: Identify Original Coordinates

First, find the coordinates of the original triangle's vertices. From the graph, \( A(0, -1) \), \( B(2, -3) \), \( C(-2, -2) \)? Wait, no, let's check the grid. Wait, the original triangle: Let's re - examine the first graph. The x - axis and y - axis: Let's assume the origin is at the center. Wait, maybe I misread. Let's take the original triangle's points: Let's say \( A(0,1) \), \( B(2,3) \), \( C(-2,2) \)? No, the first graph has the triangle with A at (0, - 1)? Wait, maybe the grid is such that the y - axis is flipped. Wait, the rotation rule for 90 degrees clockwise about the origin is \((x,y)\to(y, - x)\).

Let's correctly identify the original coordinates. Looking at the first triangle (Maria's original triangle): Let's assume the coordinates are \( A(0, - 1) \)? No, the red triangle: Let's look at the grid lines. Let's take the first triangle: Point A is at (0, - 1)? Wait, no, the y - axis has positive upwards? Wait, the first graph: The y - axis on the right has 1,2,3,4,5 upwards, and x - axis on the bottom has 1,2,3,4,5 to the right? Wait, no, the first triangle is drawn with A at (0, - 1)? Wait, maybe the coordinates are \( A(0,1) \), \( B(2,3) \), \( C(-2,2) \) (maybe the grid is labeled differently). Wait, the rotation rule for 90 degrees clockwise about the origin is \((x,y)\to(y, - x)\).

Let's take a sample point. Suppose the original point \( A(x,y) \). After 90 - degree clockwise rotation, it becomes \( (y, - x) \).

Let's look at the options. The bottom - left graph (the third one, since the first row has two, second row has two). Wait, the problem says "Maria rotated the triangle 90 degrees clockwise about the origin. What is the new triangle?"

Let's analyze the rotation rule. For a point \((x,y)\), a 90 - degree clockwise rotation about the origin transforms it to \((y, - x)\).

Let's assume the original triangle has vertices: Let's say \( A(0, - 1) \), \( B(2, - 3) \), \( C(-2, - 2) \). Applying the rotation rule \((x,y)\to(y, - x)\):

- For \( A(0, - 1) \): \((-1,0)\)
- For \( B(2, - 3) \): \((-3, - 2)\)
- For \( C(-2, - 2) \): \((-2,2)\)

Now, looking at the four graphs, the bottom - left graph (the third one, in the second row, first column) has points that match the rotated coordinates. Wait, the bottom - left graph (the one with the blue border) has points that seem to follow the 90 - degree clockwise rotation.

Alternatively, let's take the original points as \( A(0,1) \), \( B(2,3) \), \( C(-2,2) \). Rotating 90 degrees clockwise: \( A(1,0) \), \( B(3, - 2) \), \( C(2,2) \). No, that doesn't match. Wait, maybe the original triangle is in the fourth quadrant. Let's look at the four options:

The four graphs:

1. Top - left: Triangle with points in the second quadrant (negative x, positive y)
2. Top - right: Triangle with points in the second and first quadrants
3. Bottom - left: Triangle with points in the second and third quadrants (negative x, negative y)
4. Bottom - right: Triangle with points in the first and fourth quadrants

The rotation of 90 degrees clockwise about the origin: If the original triangle is in the fourth quadrant (x positive, y negative), after rotation, the points will be in the third quadrant (x negative, y negative) or second? Wait, no. Let's take a point \((x,y)\) with \( x>0,y < 0 \). After 90 - degree clockwise rotation, it becomes \((y, - x)\), where \( y<0 \) and \( - x < 0 \), so the point is in the third quadrant (x negative, y negative).

Looking at the bottom - left graph (the third one), the triangle is in the third quadrant (x negative, y negative), which matches the rotation of a fourth - quadrant triangle (original) to third - quadrant after 90 - degree clockwise rotation.

So the correct graph is the bottom - left one (the third graph, in the second row, first column).

Answer:

The bottom - left graph (the third graph when counting from top - left to bottom - right: top - left (1st), top - right (2nd), bottom - left (3rd), bottom - right (4th))