Question

a trapezoid is in the third quadrant of an x-y coordinate system, as shown here:
the trapezoid is reflected across the line y = -x, then it is reflected across the x-axis. which of these transformations will put it back in its original position (with the same orientation it had originally)?
i. a clockwise rotation of 90° about the origin
ii. a reflection across the line y = -x, then a reflection across the y-axis
iii. a reflection across the line y = x, then a reflection across the x-axis
options: i only; ii and iii only; i and ii only; i, ii and iii
question 15
1 pts
in the diagram, △a′b′c′ is the image of △abc. which type of transformation is shown?

Explanation:

Step1: Analyze the initial transformations

The trapezoid is first reflected across \( y = -x \), then across the \( x \)-axis. Let's recall the transformation rules:

• Reflection across \( y=-x \): \((x,y)\to(-y,-x)\)
• Reflection across \( x \)-axis: \((x,y)\to(x,-y)\)

Applying these to a point \((x,y)\) (initially in third quadrant, so \( x<0, y<0 \)):
First reflection: \((x,y)\to(-y,-x)\). Since \( x<0, y<0 \), \(-y>0, -x>0\) (so now in first quadrant).
Second reflection (across \( x \)-axis): \((-y,-x)\to(-y,x)\). Now \( -y>0, x<0 \) (so in fourth quadrant).

Step2: Analyze Transformation I (clockwise rotation \( 90^\circ \) about origin)

Rotation \( 90^\circ \) clockwise: \((x,y)\to(y,-x)\)
Take the point after two reflections: \((-y,x)\). Apply rotation: \((-y,x)\to(x,y)\) (which is the original point). Let's verify with coordinates:
Original point \((x,y)\) (third quadrant), after two reflections: \((-y,x)\). Rotating \( 90^\circ \) clockwise: \((x,-(-y))=(x,y)\)? Wait, no, wait: rotation \( 90^\circ \) clockwise is \((a,b)\to(b,-a)\). So for \((a,b)=(-y,x)\), it becomes \((x, -(-y))=(x,y)\). Yes, that's the original point. So I works.

Step3: Analyze Transformation II (reflect across \( y=-x \), then reflect across \( y \)-axis)

First, reflect the point after two initial reflections \((-y,x)\) across \( y=-x \): \((a,b)\to(-b,-a)\), so \((-y,x)\to(-x,y)\).
Then reflect across \( y \)-axis: \((-x,y)\to(x,y)\) (original point). Let's check:
After two initial reflections: \((-y,x)\). Reflect across \( y=-x \): \((-y,x)\to(-x,y)\) (since \( (a,b)\to(-b,-a) \)). Then reflect across \( y \)-axis: \((-x,y)\to(x,y)\). Correct. So II works.

Step4: Analyze Transformation III (reflect across \( y=x \), then reflect across \( x \)-axis)

Reflection across \( y=x \): \((x,y)\to(y,x)\)
Reflect the point after two initial reflections \((-y,x)\) across \( y=x \): \((-y,x)\to(x,-y)\)
Then reflect across \( x \)-axis: \((x,-y)\to(x,y)\) (original point). Wait, let's do it step by step:
After two initial reflections: \((-y,x)\). Reflect across \( y=x \): \((x,-y)\). Then reflect across \( x \)-axis: \((x,-y)\to(x,y)\). Correct. So III works.

Answer:

I, II and III