Question

a. reflect triangle mno over line k and label the image mno b. reflect triangle mno over line p label the image mno

Explanation:

Step1: Recall reflection rules

For reflection over a horizontal line \(y = c\), the \(x -\)coordinate of a point \((x,y)\) remains the same and the \(y -\)coordinate changes as \(y'=2c - y\). For reflection over a vertical line \(x = d\), the \(y -\)coordinate of a point \((x,y)\) remains the same and the \(x -\)coordinate changes as \(x'=2d - x\). Here, line \(k\) is \(y = 1\) and line \(p\) is \(x=- 1\).

Step2: Reflect triangle MNO over line \(k(y = 1)\)

Let the coordinates of \(M=(4,-3)\), \(N=(3,-7)\), \(O=(0,-4)\). For point \(M\), \(x\) remains \(4\), and \(y'=2\times1-(-3)=2 + 3=5\), so \(M'=(4,5)\). For point \(N\), \(x = 3\) remains the same, and \(y'=2\times1-(-7)=2 + 7 = 9\), so \(N'=(3,9)\). For point \(O\), \(x = 0\) remains the same, and \(y'=2\times1-(-4)=2 + 4=6\), so \(O'=(0,6)\).

Step3: Reflect triangle MNO over line \(p(x=-1)\)

For point \(M=(4,-3)\), \(y\) remains \(-3\), and \(x'=2\times(-1)-4=-2 - 4=-6\), so \(M''=(-6,-3)\). For point \(N=(3,-7)\), \(y=-7\) remains the same, and \(x'=2\times(-1)-3=-2 - 3=-5\), so \(N''=(-5,-7)\). For point \(O=(0,-4)\), \(y=-4\) remains the same, and \(x'=2\times(-1)-0=-2\), so \(O''=(-2,-4)\).

Answer:

a. The coordinates of \(\triangle M'N'O'\) are \(M'(4,5)\), \(N'(3,9)\), \(O'(0,6)\)
b. The coordinates of \(\triangle M''N''O''\) are \(M''(-6,-3)\), \(N''(-5,-7)\), \(O''(-2,-4)\)