Question

7 triangle rst and triangle rst are congruent. both triangles are graphed on the coordinate grid shown below. which sequence of transformations could be used to show the congruence between the triangles? a a translation 3 units up and down, then a translation 5 units to the right b a translation 2 units to the right and then a reflection across the y - axis c a reflection across the y - axis and then a translation 1 unit to the right and 3 units up d a translation 1 unit to the right and 3 units up and then a reflection across the y - axis

Explanation:

Step1: Analyze option A

A translation 3 units up and down would cancel out in the vertical direction. Then a translation 5 units to the right. But looking at the triangles, this sequence does not map one triangle to the other.

Step2: Analyze option B

A translation 2 units to the right and then a reflection across the y - axis. Let's assume a point \((x,y)\) in the original triangle. After translation 2 units to the right, it becomes \((x + 2,y)\). After reflection across the y - axis, it becomes \((-(x + 2),y)\). This does not map the original triangle to the congruent one.

Step3: Analyze option C

A reflection across the y - axis. Let a point be \((x,y)\) in the original triangle, after reflection it is \((-x,y)\). Then a translation 1 unit to the right (\((-x+1,y)\)) and 3 units up (\((-x + 1,y + 3)\)).
Let's take a vertex of the original triangle, say \(R(-6,-3)\). After reflection across the y - axis: \(R'(6,-3)\). After translation 1 unit to the right and 3 units up: \(R''(6 + 1,-3+3)=(7,0)\) (matching the corresponding vertex of the other triangle). Let's check another vertex \(S(-4,-4)\). After reflection across the y - axis: \(S'(4,-4)\). After translation 1 unit to the right and 3 units up: \(S''(4 + 1,-4 + 3)=(5,-1)\) (matching the corresponding vertex). And for \(T(-3,-1)\), after reflection across the y - axis: \(T'(3,-1)\). After translation 1 unit to the right and 3 units up: \(T''(3+1,-1 + 3)=(4,2)\) (matching the corresponding vertex).

Step4: Analyze option D

A translation 1 unit to the right and 3 units up first. Let a point \((x,y)\) become \((x + 1,y+3)\). Then a reflection across the y - axis: \((-(x + 1),y + 3)\). Taking the vertex \(R(-6,-3)\), after translation 1 unit to the right and 3 units up: \(R'(-6+1,-3 + 3)=(-5,0)\). After reflection across the y - axis: \(R''(5,0)\) which does not match the corresponding vertex of the other triangle.

Answer:

C. a reflection across the y - axis and then a translation 1 unit to the right and 3 units up