Suppose triangle TRI is rotated 270degrees 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.

Question

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Upper TUpper RUpper I

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Question content area bottom
Part 1
Given triangle Upper T prime Upper R prime Upper I prime is located in quadrant
I.
Part 2
Make a graph showing triangle TRI and triangle Upper T prime Upper R prime Upper I prime. Choose the correct answer below.
A.
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Upper TUpper RUpper IUpper T primeUpper R primeUpper I prime

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B.
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Upper TUpper RUpper IUpper T primeUpper R primeUpper I prime

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C.
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Upper TUpper RUpper IUpper T primeUpper R primeUpper I prime

Answer 1

To determine the quadrant location of triangle \( T'R'I' \) after rotating triangle \( TRI \) 270 degrees about the origin (0,0), we first need to understand the effects of the rotation on the vertices of the triangle.

Step 1: Rotating Points

The rotation of a point \( (x, y) \) around the origin by an angle \( \theta \) can be calculated using the following transformation formulas:

• Rotation by 270 degrees (or -90 degrees) results in: \[ (x', y') = (y, -x) \]

Step 2: Determine the Original Vertices

Assuming the triangle \( TRI \) has vertices at specific coordinates, let's denote:

• \( T(x_1, y_1) \)
• \( R(x_2, y_2) \)
• \( I(x_3, y_3) \)

Step 3: Apply the Transformation

Using the rotation formulas for each point of triangle \( T (x_1, y_1) \), \( R (x_2, y_2) \), and \( I (x_3, y_3) \):

• \( T' = (y_1, -x_1) \)
• \( R' = (y_2, -x_2) \)
• \( I' = (y_3, -x_3) \)

Step 4: Determine the Quadrant

After applying the 270-degree rotation:

• The x-coordinates \( y_1, y_2, y_3 \) can be positive or negative.
• The y-coordinates \( -x_1, -x_2, -x_3 \) will always be negative if \( x_1, x_2, x_3 \) are positive.

Quadrant Analysis

If \( y_1, y_2, y_3 \) are positive, then the points \( T', R', I' \) will be located in:

• Quadrant II: \( (positive, negative) \)

If \( y_1, y_2, y_3 \) are negative, then the points \( T', R', I' \) will be in:

• Quadrant III.

If vertex T was originally in Quadrant I, R was in Quadrant II, and I was in Quadrant III, the resulting triangle after 270-degree rotation would correspond to specific locations.

Final Answer

If you evaluate the original triangle's vertices based on their coordinates you can determine the location of triangle \( T'R'I' \). Assuming the original triangle's points led to a consistent quadrant designation based on this transformation:

• If triangle \( T'R'I' \) is confirmed in Quadrant II – then:

Part 2: Graphing

On your graph, you will plot points from both \( TRI \) and \( T'R'I' \) per your calculations. Each vertex from \( T, R, I \) plotted, followed by their transformed equivalents \( T', R', I' \):

• Choose from depiction options (A, B, or C) with corresponding plot configurations.

Without actual coordinates, I cannot definitively identify which graph is correct. But ensure you represent \( T, R, I \) in their respective locations, along with \( T', R', I' \) accurately according to their transformations.