Question

honors geometry 2nd nine weeks exam review
multiple choice
identify the choice that best completes the statement or answers the question.

1. point b has coordinates (-8, 1). what are the coordinates of the point when reflected across the y-axis and then rotated 180° about the origin?

a (8, 1)
b (-8, -1)
c (-8, 1)
d (8, -1)

Explanation:

Step1: Reflect over y - axis

The rule for reflecting a point \((x,y)\) across the \(y\) - axis is \((x,y)\to(-x,y)\). For the point \(B(-8,1)\), when we reflect it across the \(y\) - axis, \(x=-8\) becomes \(x = 8\) (since \(-(-8)=8\)) and \(y\) remains \(1\). So the reflected point across the \(y\) - axis is \((8,1)\).

Step2: Rotate \(180^{\circ}\) about the origin

The rule for rotating a point \((x,y)\) \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\). Now we take the point \((8,1)\) from the previous step. Applying the \(180^{\circ}\) rotation rule, \(x = 8\) becomes \(-8\) and \(y = 1\) becomes \(-1\)? Wait, no, wait. Wait, first, let's re - examine the problem. Wait, the problem says: Point \(B\) has coordinates \((-8,1)\). What are the coordinates of the point when reflected across the \(y\) - axis and then rotated \(180^{\circ}\) about the origin?

Wait, reflection across \(y\) - axis: \((x,y)\to(-x,y)\). So for \((-8,1)\), reflection across \(y\) - axis: \(x=-8\), so \(-x = 8\), \(y = 1\). So reflected point is \((8,1)\).

Now, rotation of \(180^{\circ}\) about the origin: the rule is \((x,y)\to(-x,-y)\). So for \((8,1)\), applying \(180^{\circ}\) rotation: \(x = 8\) becomes \(-8\), \(y = 1\) becomes \(-1\)? Wait, no, that's not right. Wait, no, the \(180^{\circ}\) rotation rule is that the new \(x\) - coordinate is \(-x\) and the new \(y\) - coordinate is \(-y\) of the original point (before rotation). Wait, the point after reflection is \((8,1)\). So rotating \((8,1)\) \(180^{\circ}\) about the origin: new \(x=-8\), new \(y = - 1\)? Wait, no, let's do it step by step.

Wait, maybe I made a mistake. Let's recall the rules:

• Reflection over \(y\) - axis: \((x,y)\to(-x,y)\)
• Rotation \(180^{\circ}\) about origin: \((x,y)\to(-x,-y)\)

So first, reflect \((-8,1)\) over \(y\) - axis: \((-(-8),1)=(8,1)\)

Then rotate \((8,1)\) \(180^{\circ}\) about origin: \((-8,-1)\)? Wait, no, the options don't have \((-8,-1)\). Wait, maybe the problem is: Wait, maybe I misread the problem. Let me check the options again. The options are:

A \((8,1)\)

B \((-8,-1)\)

C \((-8,1)\)

D \((8,-1)\)

Wait, maybe the problem is: Wait, the original point is \((-8,1)\). Reflect across \(y\) - axis: \((8,1)\). Then rotate \(180^{\circ}\) about origin: \((-8,-1)\)? But that's not in the options? Wait, no, maybe the problem is written incorrectly, or maybe I misread the rotation. Wait, maybe the rotation is \(180^{\circ}\) after reflection, but let's check the options again. Wait, the options are A \((8,1)\), B \((-8,-1)\), C \((-8,1)\), D \((8,-1)\). Wait, maybe the problem is: first reflect across \(y\) - axis, then rotate \(180^{\circ}\). Wait, let's recalculate.

Reflection across \(y\) - axis: \((-8,1)\to(8,1)\)

Rotation \(180^{\circ}\) about origin: \((8,1)\to(-8,-1)\), but that's option B? Wait, option B is \((-8,-1)\)? Wait, the user's options:

A \((8,1)\)

B \((-8,-1)\)

C \((-8,1)\)

D \((8,-1)\)

Wait, maybe I made a mistake in the rotation rule. Wait, the \(180^{\circ}\) rotation rule is equivalent to \((x,y)\to(-x,-y)\). So if the point after reflection is \((8,1)\), then rotating \(180^{\circ}\) gives \((-8,-1)\), which is option B. But let's check again.

Wait, maybe the problem is: Point \(B\) has coordinates \((-8,1)\). What are the coordinates of the point when reflected across the \(y\) - axis and then rotated \(180^{\circ}\) about the origin?

Step 1: Reflect across \(y\) - axis: \((x,y)\to(-x,y)\). So \((-8,1)\to(8,1)\)

Step 2: Rotate \(180^{\circ}\) about origin: \((x,y)\to(-x,-y)\). So \((8,1)\to(-8,-1)\), which is…

Answer:

B. \((-8,-1)\)