Question

which best explains whether or not triangles rst and acb are congruent? the figures are congruent. △rst can be mapped to △acb by a reflection over the x - axis and a translation 2 units to the left. the figures are congruent. △rst can be mapped to △acb by a reflection over the y - axis and a translation 2 units down. the figures are not congruent. point r corresponds to point a, but s corresponds to b and t corresponds to c. the figures are not congruent. point r does not correspond with point a.

Explanation:

Step1: Analyze reflection and translation

• Reflection over the \(y\) - axis: \((x,y)\to(-x,y)\).
• Translation 2 units down: \((x,y)\to(x,y - 2)\)
• For \(\triangle RST\) with \(R(1,1)\), \(S(1,5)\), \(T(4,1)\)
• After reflection over \(y\) - axis: \(R(- 1,1)\), \(S(-1,5)\), \(T(-4,1)\)
• After translation 2 units down: \(R(-1,-1)\), \(S(-1,3)\), \(T(-4,-1)\)
• For \(\triangle ACB\) with \(A(-1,-1)\), \(B(2,-1)\), \(C(-1,-5)\) (Note: assume correct correspondence of vertices after transformation, lengths of sides of \(\triangle RST\): \(RS = 4\), \(RT=3\), \(ST = 5\) (by distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(RS=\sqrt{(1 - 1)^2+(5 - 1)^2}=4\), \(RT=\sqrt{(4 - 1)^2+(1 - 1)^2}=3\), \(ST=\sqrt{(4 - 1)^2+(1 - 5)^2}=5\); for \(\triangle ACB\), \(AC = 4\), \(AB = 3\), \(BC = 5\))

Step2: Check congruence criteria

• Congruent triangles have all corresponding sides equal (SSS - Side - Side - Side criterion). Since reflection and translation are rigid motions (preserve shape and size), and we can map \(\triangle RST\) to \(\triangle ACB\) by reflection over \(y\) - axis and translation 2 units down.

Answer:

The figures are congruent. \(\triangle RST\) can be mapped to \(\triangle ACB\) by a reflection over the \(y\) - axis and a translation 2 units down.