Question

the graph shows triangles qrs and xyz.
is qrs congruent to xyz? justify your answer.
yes, because a reflection across the x - axis maps qrs onto xyz.
yes, because a translation right 2 units and up 9 units maps qrs onto xyz.
no, because ∠r and ∠y do not have the same measure.
no, because (overline{qr}) and (overline{xy}) do not have the same length.

Explanation:

Step1: Calculate the length of \( \overline{QR} \)

Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Let \( Q=(5,-3) \) and \( R=(7,-4) \).

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Step2: Calculate the length of \( \overline{XY} \)

Let \( X=(6,6) \) and \( Y=(8,5) \).

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Step3: Check the translation

For a translation, if we consider the \( x \)-coordinates: \( 5+1 = 6\), \(7 + 1=8\) (not \(+2\)). For the \( y \)-coordinates: \(-3+9=6\), \(-4 + 9=5\). A translation right \(1\) unit and up \(9\) units maps \( Q=(5,-3)\) to \(X=(6,6)\) and \(R=(7,-4)\) to \(Y=(8,5)\) and \(S=(4,-8)\) to \(Z=(5,2)\).

Answer:

Yes, because a translation right \(1\) unit and up \(9\) units maps \(QRS\) onto \(XYZ\).