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A coordinate plane with 4 quadrants shows x and y axes ranging from negative 9 to 9 in increments of 1. Three rectangles are formed by lines joining four plotted points each. The coordinates of the plotted points for first rectangle joined by solid lines are left parenthesis negative 5 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 6 right parenthesis, and left parenthesis negative 5 comma negative 6 right parenthesis. The coordinates for the second rectangle joined by lines made of dashes and dots are left parenthesis 3 comma 4 right parenthesis, left parenthesis 5 comma 4 right parenthesis, left parenthesis 5 comma 8 right parenthesis, and left parenthesis 3 comma 8 right parenthesis. The coordinates of the plotted points for the third rectangle joined by dotted lines are left parenthesis negative 4 comma 1 right parenthesis, left parenthesis negative 6 comma 1 right parenthesis, left parenthesis negative 6 comma 5 right parenthesis, and left parenthesis negative 4 comma 5 right parenthesis.
The first step to determine that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I was to rotate the first rectangle 90 degrees clockwise about the origin. After rotating the first rectangle, the next step is to compare the dimensions and positions of the rectangles to establish their congruence. So what is the answer?

1 answer

To determine if the rectangles in Quadrant III (the first rectangle) and Quadrant I (the second rectangle) are congruent, we follow the steps:

  1. Rotation of the First Rectangle: The coordinates of the first rectangle are \((-5, -4)\), \((-1, -4)\), \((-1, -6)\), and \((-5, -6)\). Rotating these points 90 degrees clockwise around the origin, we obtain:

    • \((-5, -4) \rightarrow (-4, 5)\)
    • \((-1, -4) \rightarrow (-4, 1)\)
    • \((-1, -6) \rightarrow (-6, 1)\)
    • \((-5, -6) \rightarrow (-6, 5)\) After rotation, the new coordinates for the rectangle become \((-4, 5)\), \((-6, 5)\), \((-6, 1)\), \((-4, 1)\).

  2. Dimensions of the Rectangles:

    • The first rectangle (original) has a width of \(4\) units (from \(-5\) to \(-1\)) and a height of \(2\) units (from \(-6\) to \(-4\)).
    • The second rectangle has a width of \(2\) units (from \(3\) to \(5\)) and a height of \(4\) units (from \(4\) to \(8\)).

  3. Comparison: After rotation, the dimensions of the rotated first rectangle are \(2\) units wide and \(4\) units high (consistent with the second rectangle's dimensions, which are \(2\) units wide and \(4\) units high respectively).

Thus, since the dimensions are equal after the rotation and verification, the two rectangles in Quadrant I and Quadrant III are congruent.

Final Answer: The rectangles are congruent.