Question

a line segment with endpoints $(7,5)$ and $(10,1)$ is reflected across the line $y = -x$. which pair of coordinates represents the endpoints of the given segment after the reflection?
a. $(-7,5)$ and $(-10,1)$
b. $(-5,-7)$ and $(-1,-10)$
c. $(-5,7)$ and $(-1,10)$
d. $(7,-5)$ and $(10,-1)$

Explanation:

Step1: Recall the reflection rule over \(y = -x\)

When a point \((x,y)\) is reflected over the line \(y=-x\), the transformation rule is \((x,y)\to(-y,-x)\).

Step2: Apply the rule to the first endpoint \((7,5)\)

For the point \((7,5)\), using the rule \((x,y)\to(-y,-x)\), we substitute \(x = 7\) and \(y = 5\). Then \(-y=-5\) and \(-x=-7\), so the new point is \((-5,-7)\).

Step3: Apply the rule to the second endpoint \((10,1)\)

For the point \((10,1)\), using the rule \((x,y)\to(-y,-x)\), we substitute \(x = 10\) and \(y = 1\). Then \(-y=-1\) and \(-x=-10\), so the new point is \((-1,-10)\).

Answer:

B. \((-5,-7)\) and \((-1,-10)\)