Question

graph \\( \triangle j k l \\) and its image after a reflection in the line \\( y = - 3 \\).
\\( j ( 3, - 5 ), k ( 4, - 1 ), l ( 0, - 3 ) \\)
12

Explanation:

Step1: Find the distance from each point to the line \(y = - 3\)

For a point \((x,y)\), the distance \(d\) from the point to the line \(y=-3\) is \(d=\vert y - (-3)\vert=\vert y + 3\vert\).

• For \(J(3,-5)\): \(d_J=\vert-5+3\vert = 2\)
• For \(K(4,-1)\): \(d_K=\vert-1 + 3\vert=2\)
• For \(L(0,-3)\): \(d_L=\vert-3+3\vert = 0\)

Step2: Reflect each point over the line \(y=-3\)

When reflecting a point \((x,y)\) over the line \(y = a\), the formula for the \(y\)-coordinate of the new point \((x,y')\) is \(y'=2a - y\), and the \(x\)-coordinate remains the same. Here \(a=-3\).

• For \(J(3,-5)\): \(y'_J=2\times(-3)-(-5)=-6 + 5=-1\), so \(J'(3,-1)\)
• For \(K(4,-1)\): \(y'_K=2\times(-3)-(-1)=-6 + 1=-5\), so \(K'(4,-5)\)
• For \(L(0,-3)\): \(y'_L=2\times(-3)-(-3)=-6+3=-3\), so \(L'(0,-3)\)

Step3: Graph the original and reflected triangles

Plot the points \(J(3,-5)\), \(K(4,-1)\), \(L(0,-3)\) for \(\triangle JKL\) and \(J'(3,-1)\), \(K'(4,-5)\), \(L'(0,-3)\) for the reflected \(\triangle J'K'L'\) on the coordinate - plane.

Answer:

The coordinates of the reflected triangle \(\triangle J'K'L'\) are \(J'(3,-1)\), \(K'(4,-5)\), \(L'(0,-3)\). Graph \(\triangle JKL\) with vertices \(J(3,-5)\), \(K(4,-1)\), \(L(0,-3)\) and \(\triangle J'K'L'\) with vertices \(J'(3,-1)\), \(K'(4,-5)\), \(L'(0,-3)\) on the given coordinate grid.