Math Problem Statement

Rotate ∆PQR 270° counterclockwise around the origin.

Solution

The problem shown in the image asks to rotate triangle PQR\triangle PQR by 270° counterclockwise around the origin. Let's break down how to approach this type of transformation:

  1. Rotation of points 270° counterclockwise:

    • The general rule for a 270° counterclockwise rotation around the origin is to switch the coordinates of each point and negate the new first coordinate.
    • For a point (x,y)(x, y), the transformation is: (x,y)(y,x)(x, y) \rightarrow (y, -x)
  2. Applying the rule to each vertex of the triangle:

    • Assume the coordinates of PP, QQ, and RR are P(x1,y1)P(x_1, y_1), Q(x2,y2)Q(x_2, y_2), and R(x3,y3)R(x_3, y_3). Based on the diagram, these points will be rotated using the rule (x,y)(y,x)(x, y) \rightarrow (y, -x).
    • After the transformation, the new coordinates of PP', QQ', and RR' will be plotted on the graph.

Would you like me to calculate the new coordinates of the points PP, QQ, and RR from the diagram? Let me know if you need more details on any specific part of the process!

Further Questions to Explore:

  1. What is the rule for a 90° counterclockwise rotation around the origin?
  2. How does the rotation affect the orientation of the shape?
  3. What happens to a shape's size and angles during rotation?
  4. How can you verify the accuracy of a transformation on a graph?
  5. How does the triangle's image differ after a reflection instead of a rotation?

Tip: Always remember the origin is the fixed point during any rotation unless stated otherwise.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Transformations
Rotations

Formulas

Rotation Formula for 270°: (x, y) → (y, -x)

Theorems

Rotation Theorem around Origin

Suitable Grade Level

Grades 7-9