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 \(y = k\), the formula for the reflection is \((x,2k - y)\). Here \(k=-3\), so the formula is \((x,-6 - y)\)

• For \(J(3,-5)\): \(J'(3,-6-(-5))=J'(3,-1)\)
• For \(K(4,-1)\): \(K'(4,-6-(-1))=K'(4,-5)\)
• For \(L(0,-3)\): \(L'(0,-6-(-3))=L'(0,-3)\)

Step3: Graph \(\triangle JKL\) and \(\triangle J'K'L'\)

• Plot \(J(3,-5)\), \(K(4,-1)\), \(L(0,-3)\) and connect them to form \(\triangle JKL\)
• Plot \(J'(3,-1)\), \(K'(4,-5)\), \(L'(0,-3)\) and connect them to form \(\triangle J'K'L'\)

Answer:

The coordinates of the image of \(\triangle JKL\) after reflection over the line \(y = - 3\) are \(J'(3,-1)\), \(K'(4,-5)\), \(L'(0,-3)\). Graph \(\triangle JKL\) with vertices \((3,-5)\), \((4,-1)\), \((0,-3)\) and \(\triangle J'K'L'\) with vertices \((3,-1)\), \((4,-5)\), \((0,-3)\) on the coordinate - plane.