Question

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which statement best explains whether \\( \triangle j k l \\) is
congruent to \\( \triangle m n p \\)
\\( \triangle j k l \\) is congruent to \\( \triangle m n p \\) because
\\( \triangle j k l \\) can be mapped to \\( \triangle m n p \\) by a
reflection across the \\( x \\)-axis followed by a
rotation of \\( 180^{\circ} \\) about the origin
\\( \triangle j k l \\) is not congruent to \\( \triangle m n p \\)
because there is no sequence of rigid motions
that maps \\( \triangle j k l \\) to \\( \triangle m n p \\)
\\( \triangle j k l \\) is congruent to \\( \triangle m n p \\) because
\\( \triangle j k l \\) can be mapped to \\( \triangle m n p \\) by a
reflection across the \\( y \\)-axis
\\( \triangle j k l \\) is congruent to \\( \triangle m n p \\) because
\\( \triangle j k l \\) can be mapped to \\( \triangle m n p \\) by a

Explanation:

Step1: Recall the properties of rigid motions

Rigid motions (reflections, rotations, translations) preserve the shape and size of a figure. If two triangles are congruent, there exists a sequence of rigid motions that maps one to the other.

Step2: Analyze the first option

A reflection across the \(x -\)axis followed by a \(180^{\circ}\) rotation about the origin. Let's assume a general point \((x,y)\) in \(\triangle JKL\). A reflection across the \(x -\)axis gives \((x, - y)\), and then a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\) (for the general rule \((x,y)\to(-x,-y)\) after \(180^{\circ}\) rotation). This will not map \(\triangle JKL\) to \(\triangle MNP\) as the orientation and position do not match.

Step3: Analyze the second option

Check if there is a sequence of rigid motions. Let's use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(\triangle JKL\):
Let \(J(-4,2)\), \(K(-2,3)\), \(L(-2, - 1)\)
\(JK=\sqrt{(-2 + 4)^2+(3 - 2)^2}=\sqrt{4 + 1}=\sqrt{5}\)
\(KL=\sqrt{(-2+2)^2+( - 1 - 3)^2}=4\)
\(JL=\sqrt{(-2 + 4)^2+( - 1 - 2)^2}=\sqrt{4 + 9}=\sqrt{13}\)

For \(\triangle MNP\):
Let \(M(4,0)\), \(P(2,3)\), \(N(2,-1)\)
\(MP=\sqrt{(4 - 2)^2+(0 - 3)^2}=\sqrt{4 + 9}=\sqrt{13}\)
\(PN=\sqrt{(2 - 2)^2+( - 1 - 3)^2}=4\)
\(MN=\sqrt{(4 - 2)^2+(0 + 1)^2}=\sqrt{4+1}=\sqrt{5}\)

Now, check the mapping. A reflection across the \(y -\)axis: the rule is \((x,y)\to(-x,y)\). If we reflect \(\triangle JKL\) (with vertices \(J(-4,2)\), \(K(-2,3)\), \(L(-2,-1)\)) across the \(y -\)axis, we get \(J'(4,2)\), \(K'(2,3)\), \(L'(2,-1)\). Then we can see that \(\triangle J'K'L'\) (after reflection) can be translated (in this case, since the \(y -\)coordinates of \(P\) and \(K'\) are the same, and \(x -\)coordinates of \(M\) and \(J'\) are related, \(\triangle JKL\) can be mapped to \(\triangle MNP\) by a reflection across the \(y -\)axis.

Answer:

\(\triangle JKL\) is congruent to \(\triangle MNP\) because \(\triangle JKL\) can be mapped to \(\triangle MNP\) by a reflection across the \(y -\)axis.