Question

is each pair of triangles congruent? justify your answer using rigid transformations.

1. △jkl and △rst
2. △abc and △xyz

Explanation:

Step1: Check for rigid transformations (translation, rotation, reflection)

Rigid transformations preserve shape and size. For \(\triangle JKL\) and \(\triangle RST\), we can observe that \(\triangle RST\) can be rotated and translated to match \(\triangle JKL\).

Step2: Verify side - length and angle - measure preservation

By counting the grid units (assuming each grid square has side - length \(1\)), we can find the lengths of the sides of the triangles.
For \(\triangle JKL\):

• \(JK = 3\) (vertical distance from \(J\) to \(K\)), \(KL=\sqrt{(3 - 0)^{2}+(0 - 3)^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\), \(JL=\sqrt{(3 - 0)^{2}+(0+ 3)^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\)

For \(\triangle RST\):

• \(RS = 3\) (vertical distance from \(R\) to \(S\)), \(ST=\sqrt{(0+3)^{2}+(3 - 0)^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\), \(RT=\sqrt{(0 + 3)^{2}+(0 - 3)^{2}}=\sqrt{9+9}=\sqrt{18}=3\sqrt{2}\)

Since rigid transformations (a rotation about the origin by \(180^{\circ}\) followed by a translation) can map \(\triangle RST\) to \(\triangle JKL\) and all corresponding side - lengths are equal (\(RS = JK\), \(ST = KL\), \(RT = JL\)), the triangles are congruent.

For \(\triangle ABC\) and \(\triangle XYZ\):

Step1: Check for rigid - transformation mapping

We assume a translation vector \((x,y)\). Let the coordinates of the vertices (assuming \(A=(x_1,y_1)\), \(B=(x_2,y_2)\), \(C=(x_3,y_3)\), \(X=(x_4,y_4)\), \(Y=(x_5,y_5)\), \(Z=(x_6,y_6)\))
If we consider the side - lengths:

• Using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)

For \(\triangle ABC\):
Let \(A=( - 4,-2)\), \(B=(0,-1)\), \(C=( - 3,-4)\)
\(AB=\sqrt{(0 + 4)^{2}+(-1 + 2)^{2}}=\sqrt{16 + 1}=\sqrt{17}\), \(BC=\sqrt{( - 3-0)^{2}+(-4 + 1)^{2}}=\sqrt{9 + 9}=\sqrt{18}\), \(AC=\sqrt{( - 3 + 4)^{2}+(-4 + 2)^{2}}=\sqrt{1+4}=\sqrt{5}\)
For \(\triangle XYZ\):
Let \(X=( - 4,0)\), \(Y=(1,1)\), \(Z=( - 5,3)\)
\(XY=\sqrt{(1 + 4)^{2}+(1 - 0)^{2}}=\sqrt{25 + 1}=\sqrt{26}\), \(YZ=\sqrt{( - 5-1)^{2}+(3 - 1)^{2}}=\sqrt{36 + 4}=\sqrt{40}\), \(XZ=\sqrt{( - 5 + 4)^{2}+(3 - 0)^{2}}=\sqrt{1 + 9}=\sqrt{10}\)

Since there is no rigid transformation (translation, rotation, reflection) that can map \(\triangle XYZ\) to \(\triangle ABC\) (because corresponding side - lengths are not equal), the triangles are not congruent.

Answer:

1. \(\triangle JKL\) and \(\triangle RST\) are congruent. A rotation about the origin by \(180^{\circ}\) followed by a translation can map \(\triangle RST\) to \(\triangle JKL\), and all corresponding side - lengths are equal.
2. \(\triangle ABC\) and \(\triangle XYZ\) are not congruent. There is no rigid transformation that can map one triangle to the other as their corresponding side - lengths (calculated using the distance formula \(d = \sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)) are not equal.