Question

△fgh is reflected across the x - axis. what are the signs of the coordinates of the vertices of △fgh? positive x - coordinates positive y - coordinates positive x - coordinates negative y - coordinates negative x - coordinates positive y - coordinates negative x - coordinates negative y - coordinates

Explanation:

Step1: Recall the reflection rule

When a point \((x,y)\) is reflected across the \(x -\)axis, the transformation rule is \((x,y)\to(x, - y)\).

Step2: Analyze the \(x -\)coordinate sign

Looking at the original triangle \(\triangle FGH\), all its vertices \(F\), \(G\), \(H\) have negative \(x -\)coordinates (since they are to the left of the \(y -\)axis). After reflection across the \(x -\)axis, the \(x -\)coordinate of each vertex \((x,y)\to(x, - y)\) remains the same. So the \(x -\)coordinates of \(\triangle F'G'H'\) are still negative.

Step3: Analyze the \(y -\)coordinate sign

The original triangle \(\triangle FGH\) has vertices with negative \(y -\)coordinates (since they are below the \(x -\)axis). Using the reflection rule \((x,y)\to(x, - y)\), if \(y<0\), then \(-y>0\) is false. Wait, no, wait the original \(y\) of \(\triangle FGH\) is negative. After reflection \((x,y)\to(x, - y)\), if \(y=-k,k > 0\), then \(-y = k>0\) is wrong. Wait no:
The original triangle \(\triangle FGH\) has vertices with \(y<0\). After reflection across the \(x -\)axis (transformation \((x,y)\to(x,-y)\)), if \(y\) is negative (say \(y=-m,m>0\)), then \(-y=m > 0\) is wrong. Wait no:
The original \(\triangle FGH\) has vertices with \(y<0\). After reflection \((x,y)\to(x, - y)\), if \(y=-a,a>0\), then \(-y=a>0\) is wrong. Wait, no:
The rule for reflection over the \(x -\)axis is \((x,y)\to(x,-y)\). The original \(\triangle FGH\) has \(x<0\) (left of \(y -\)axis) and \(y < 0\) (below \(x -\)axis). After reflection \((x,y)\to(x,-y)\), the \(x\) remains negative (because reflection over \(x -\)axis does not change the \(x\) - value for the sign related to left - right position), and \(y\) (originally negative) becomes positive.

Answer:

Negative \(x -\)coordinates, Positive \(y -\)coordinates.