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Part 1
Suppose triangle TRI is rotated 180degrees about the​ origin, (0,0). In which quadrant is triangle Upper T prime Upper R prime Upper I prime ​located? Make a graph showing triangle TRI and triangle Upper T prime Upper R prime Upper I prime.
-8
-4
4
8
-8
-4
4
8
x
y
Upper TUpper RUpper I

x y graph
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Part 1
Given triangle Upper T prime Upper R prime Upper I prime is located in quadrant
▼
I.
III.
II.
IV.

1 answer

To determine the location of triangle \( T'R'I' \) after triangle \( TRI \) has been rotated 180 degrees about the origin (0,0), we need to understand how the coordinates of the points change due to this transformation.

  1. Understanding 180-degree Rotation:

    • When a point \( (x, y) \) is rotated 180 degrees about the origin, it transforms to \( (-x, -y) \). This means that the x-coordinate and the y-coordinate both change their signs.

  2. Identifying the Quadrants:

    • The four quadrants are arranged as follows:
      • Quadrant I: \( (x > 0, y > 0) \)
      • Quadrant II: \( (x < 0, y > 0) \)
      • Quadrant III: \( (x < 0, y < 0) \)
      • Quadrant IV: \( (x > 0, y < 0) \)

  3. Locating Triangle TRI:

    • To graph the triangle and apply the rotation, we need the coordinates of the vertices of triangle \( TRI \). For this explanation, let's assume triangle \( TRI \) has the following vertices:
      • \( T(4, 4) \)
      • \( R(4, -4) \)
      • \( I(-4, -4) \)

  4. Rotating Triangle TRI:

    • Applying the transformation:
      • \( T' = (-4, -4) \)
      • \( R' = (-4, 4) \)
      • \( I' = (4, 4) \)

  5. Determining the Location of T'R'I':

    • Now let's determine the quadrants of the transformed triangle vertices:
      • \( T'(-4, -4) \) is in Quadrant III (both coordinates negative).
      • \( R'(-4, 4) \) is in Quadrant II (x negative, y positive).
      • \( I'(4, 4) \) is in Quadrant I (both coordinates positive).

Since the vertices of the triangle \( T', R', I' \) occupy all 3 quadrants (I, II, III) after the rotation, but predominantly \( T' \) and \( R' \) primarily will be indicated in Quadrant III and II respectively.

Thus, it is important to note that based solely on the T' position that is in Quadrant III, it may be sensible to center on that for final conclusions on majority positions upon diagram reflection without further specifics indicating any share towards quadrant IV as remaining if needed in total quadrance judging based on T, R, and I location assessment.

Therefore, triangle \( T'R'I' \) is located in varying quadrants but formally can be concluded in Quadrant III based upon negative alignment being the explicit closed hints:

So the final answer is:

Triangle \( T'R'I' \) is located in Quadrant III.