Question

review college mod 4 test
date

1. select the graph that represents △ pqr with vertices p(1, 3), q(3, −2) and r(4, 2) and its image after a rotation 90° counterclockwise about the origin.

the rule for a 90° counterwise rotation about origin is (x,y)→(-y,x)
for example: (c-3,1), q(2,3), r(-2,4)

Explanation:

Step1: Recall rotation rule

For a 90° counterclockwise rotation about the origin, the rule is \((x, y) \to (-y, x)\).

Step2: Apply rule to \(P(1, 3)\)

Substitute \(x = 1\), \(y = 3\) into the rule: \((-3, 1)\), so \(P'(-3, 1)\).

Step3: Apply rule to \(Q(3, -2)\)

Substitute \(x = 3\), \(y = -2\) into the rule: \((2, 3)\), so \(Q'(2, 3)\).

Step4: Apply rule to \(R(4, 2)\)

Substitute \(x = 4\), \(y = 2\) into the rule: \((-2, 4)\), so \(R'(-2, 4)\).

Now, we check the graphs. The original triangle \(PQR\) has vertices \(P(1,3)\), \(Q(3,-2)\), \(R(4,2)\). The image \(P'Q'R'\) should have vertices \(P'(-3,1)\), \(Q'(2,3)\), \(R'(-2,4)\). Looking at the graphs, the one where the original triangle is blue with \(P\) in the first quadrant, \(Q\) below the x - axis, \(R\) in the first quadrant, and the image is green with \(P'\) in the second quadrant, \(Q'\) in the first quadrant, \(R'\) in the second quadrant (matching the calculated coordinates) is the correct graph. From the given options, the graph (the bottom - left one? Wait, no, let's re - check. Wait, the first graph on the left top: Wait, no, let's re - evaluate. Wait, the correct image after rotation: \(P(1,3)\) becomes \(P'(-3,1)\), \(Q(3,-2)\) becomes \(Q'(2,3)\), \(R(4,2)\) becomes \(R'(-2,4)\). So the original triangle has \(P\) at (1,3), \(Q\) at (3, - 2), \(R\) at (4,2). The image has \(P'\) at (- 3,1), \(Q'\) at (2,3), \(R'\) at (- 2,4). So looking at the graphs, the graph where the original is blue with \(P\) (1,3), \(Q\) (3, - 2), \(R\) (4,2) and the image is green with \(P'\) (- 3,1), \(Q'\) (2,3), \(R'\) (- 2,4) is the correct one. From the given figures, the graph (the one with original \(P\) at (1,3), \(Q\) at (3, - 2), \(R\) at (4,2) and image \(P'\) at (- 3,1), \(Q'\) at (2,3), \(R'\) at (- 2,4)) is the correct representation.

Answer:

The graph (identify the correct one from the given options, for example, if we consider the options as per the image: the graph with original \(\triangle PQR\) (blue) with \(P(1,3)\), \(Q(3, - 2)\), \(R(4,2)\) and image \(\triangle P'Q'R'\) (green) with \(P'(-3,1)\), \(Q'(2,3)\), \(R'(-2,4)\) (the graph that matches these coordinates)).