Question

1. select the graph that represents $\triangle pqr$ with vertices $p(1, 3)$, $q(3, -2)$ and $r(4, 2)$ and its image after a rotation $90\degree$ counterclockwise about the origin.

Explanation:

Step1: Recall rotation rule

The rule for a \(90^\circ\) counterclockwise rotation about the origin is \((x,y)\to(-y,x)\).

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

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

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

For \(Q(3,-2)\), \(x = 3,y=-2\), so \(Q'=(2,3)\) (since \(-y = -(-2)=2\), \(x = 3\)).

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

For \(R(4,2)\), \(x = 4,y = 2\), so \(R'=(-2,4)\).

Now, check the first graph: \(P'\) is \((-3,1)\), \(Q'\) is \((2,3)\), \(R'\) is \((-2,4)\) – which matches the transformed points. The second graph's transformed points don't match the rotation rule results.

Answer:

The first graph (the one with \(P(1,3)\), \(Q(3,-2)\), \(R(4,2)\) in blue and \(P'\), \(Q'\), \(R'\) in green)