Question

△fgh is rotated 90° counterclockwise around the origin to form △fgh.

1. which coordinate is the same as the

x-coordinate of point g?
opposite of y-coordinate of point g

1. which coordinate is the same as the

y-coordinate of point g?
☑️ x-coordinate of point g
☑️ y-coordinate of point g
☑️ opposite of x-coordinate of point g
☑️ opposite of y-coordinate of point g
(image shows coordinate plane with △fgh and △fgh, points: f, g(-2, -4), h; f, g(4, -2), h)

Explanation:

Step1: Recall 90° counterclockwise rotation rule

The rule for rotating a point \((x, y)\) 90° counterclockwise around the origin is \((x, y) \to (-y, x)\).

Step2: Identify coordinates of \(G\)

From the graph, point \(G\) has coordinates \((-2, -4)\), so \(x = -2\) and \(y = -4\).

Step3: Apply rotation rule to \(G\)

Using the rule \((x, y) \to (-y, x)\), substitute \(x = -2\) and \(y = -4\). We get \(-y = -(-4)=4\) and \(x = -2\)? Wait, no, wait: Wait, the rotation rule is \((x,y)\) becomes \((-y, x)\). Wait, \(G\) is \((-2, -4)\), so \(x=-2\), \(y = -4\). Then \(-y = -(-4)=4\), and \(x=-2\)? Wait, no, the rotated point \(G'\) is \((4, -2)\). Let's check: For \(G(-2, -4)\), applying 90° counterclockwise rotation: the new \(x\)-coordinate is \(-y\) (since original \(y\) is \(-4\), \(-y = 4\)), and the new \(y\)-coordinate is \(x\) (original \(x\) is \(-2\), so \(y\)-coordinate of \(G'\) is \(-2\)). Wait, the question is about the \(y\)-coordinate of \(G'\). The \(y\)-coordinate of \(G'\) is \(-2\). Now, let's look at the original \(G(-2, -4)\). The options are about which coordinate of \(G\) (original) matches the \(y\)-coordinate of \(G'\) (\(-2\)). Let's check each option:

• Option 1: \(x\)-coordinate of \(G\) is \(-2\). No, \(-2\) vs \(-2\)? Wait, \(G'\) has \(y\)-coordinate \(-2\), and original \(G\) has \(x\)-coordinate \(-2\)? Wait, no, wait the rotation rule: \((x,y)\) rotated 90° counterclockwise is \((-y, x)\). So for \(G(-2, -4)\), \(x=-2\), \(y=-4\). So new \(x\) is \(-y = -(-4)=4\), new \(y\) is \(x = -2\). So \(G'\) is \((4, -2)\). So the \(y\)-coordinate of \(G'\) is \(-2\), which is equal to the \(x\)-coordinate of \(G\) (which is \(-2\))? Wait, no, the \(x\)-coordinate of \(G\) is \(-2\), and the \(y\)-coordinate of \(G'\) is \(-2\). Wait, but let's check the options. Wait, the options are:
• \(x\)-coordinate of point \(G\): \(-2\)
• \(y\)-coordinate of point \(G\): \(-4\)
• Opposite of \(x\)-coordinate of point \(G\): \(2\)
• Opposite of \(y\)-coordinate of point \(G\): \(4\)

The \(y\)-coordinate of \(G'\) is \(-2\), which is equal to the \(x\)-coordinate of \(G\) (\(-2\))? Wait, no, wait the \(y\)-coordinate of \(G'\) is \(-2\), and the \(x\)-coordinate of \(G\) is \(-2\). Wait, but let's re-express: The \(y\)-coordinate of \(G'\) is \(-2\). The original \(G\) has \(x\)-coordinate \(-2\), \(y\)-coordinate \(-4\). So the \(y\)-coordinate of \(G'\) is \(-2\), which is equal to the \(x\)-coordinate of \(G\) (\(-2\))? Wait, no, the \(x\)-coordinate of \(G\) is \(-2\), and the \(y\)-coordinate of \(G'\) is \(-2\). So the \(y\)-coordinate of \(G'\) is equal to the \(x\)-coordinate of \(G\)? Wait, no, the rotation rule says that the \(y\)-coordinate of the rotated point is the original \(x\)-coordinate. Because \((x,y)\) rotated 90° counterclockwise is \((-y, x)\), so the new \(y\)-coordinate is \(x\) (the original \(x\)-coordinate). So the \(y\)-coordinate of \(G'\) is equal to the \(x\)-coordinate of \(G\). Let's confirm with \(G(-2, -4)\): original \(x = -2\), so new \(y\) is \(x = -2\), which matches \(G'\)’s \(y\)-coordinate \(-2\). So the correct option is "x-coordinate of point G". Wait, but let's check the options again. The options are:

• \(x\)-coordinate of point \(G\)
• \(y\)-coordinate of point \(G\)
• Opposite of \(x\)-coordinate of point \(G\)
• Opposite of \(y\)-coordinate of point \(G\)

So the \(y\)-coordinate of \(G'\) is \(-2\), which is the \(x\)-coordinate of \(G\) (since \(G\) is \((-2, -4)\), \(x\)-coordinate is \(-2\)). So the correct option is "x-coordinate of point G".

Answer:

A. \(x\)-coordinate of point \(G\) (assuming the first option is labeled as A, but based on the options given, the correct one is "x-coordinate of point G" among the options provided. Wait, the options are:

1. \(x\)-coordinate of point \(G\)

1. \(y\)-coordinate of point \(G\)

1. Opposite of \(x\)-coordinate of point \(G\)

1. Opposite of \(y\)-coordinate of point \(G\)

So the correct answer is the first option: \(x\)-coordinate of point \(G\)