Question

1. rotation 180° about the origin

Explanation:

Step1: Identify coordinates of points

First, we find the coordinates of points \( U \), \( T \), and \( V \) from the grid.

• Let's assume each grid square has side length 1.
• For point \( U \): Looking at the grid, it is at \( (-3, 1) \) (3 units left of origin on x - axis, 1 unit up on y - axis).
• For point \( T \): It is at \( (0, - 1) \) (on y - axis, 1 unit down from origin).
• For point \( V \): It is at \( (-2, - 4) \) (2 units left of origin on x - axis, 4 units down on y - axis).

Step2: Apply 180° rotation rule

The rule for rotating a point \( (x,y) \) 180° about the origin is \( (x,y)\to(-x,-y) \).

• For point \( U(-3,1) \):

Applying the rule, \( x=-3\), \( y = 1 \), so the new coordinates \( U' \) are \( -(-3)=3 \), \( -1=-1 \), so \( U'(3, - 1) \).

• For point \( T(0,-1) \):

Applying the rule, \( x = 0\), \( y=-1 \), so the new coordinates \( T' \) are \( -0 = 0 \), \( -(-1)=1 \), so \( T'(0,1) \).

• For point \( V(-2,-4) \):

Applying the rule, \( x=-2\), \( y = - 4 \), so the new coordinates \( V' \) are \( -(-2)=2 \), \( -(-4)=4 \), so \( V'(2,4) \).

Step3: Plot the new points (optional for understanding)

To visualize the rotated triangle, we can plot the points \( U'(3, - 1) \), \( T'(0,1) \), and \( V'(2,4) \) on the coordinate plane. The rotated triangle \( U'T'V' \) is the image of triangle \( UTV \) after a 180° rotation about the origin.

Answer:

The coordinates of the rotated points are \( U'(3, - 1) \), \( T'(0,1) \), and \( V'(2,4) \). The rotated triangle is formed by these points. (If the question was to find the coordinates of the image after rotation, this is the result. If it was to draw, the process of plotting the points as above gives the rotated figure.)