Question

triangle rst is rotated 180° about the origin, and then translated up 3 units. which congruency statement describes the figures?

△rst ≅ △acb
△rst ≅ △abc
△rst ≅ △bca
△rst ≅ △bac

Explanation:

Step1: Understand the properties of rotation and translation

Rotation and translation are rigid transformations. Rigid transformations preserve the shape and size of the figure, so the pre - image and the image are congruent. We just need to match the corresponding vertices.

Step2: Analyze the corresponding vertices

When \(\triangle RST\) is rotated \(180^{\circ}\) about the origin and translated up \(3\) units:

• Vertex \(R(1,1)\): After rotation \(180^{\circ}\) about the origin \((x,y)\to(-x, - y)\), \(R(1,1)\to(-1,-1)\), then after translation up \(3\) units \((-1,-1)\to(-1,2)\) which corresponds to vertex \(B\).
• Vertex \(S(3,4)\): After rotation \(180^{\circ}\) about the origin \((3,4)\to(-3,-4)\), then after translation up \(3\) units \((-3,-4)\to(-3,-1)\) which corresponds to vertex \(A\).
• Vertex \(T(5,0)\): After rotation \(180^{\circ}\) about the origin \((5,0)\to(-5,0)\), then after translation up \(3\) units \((-5,0)\to(-5,3)\) which corresponds to vertex \(C\).

So \(\triangle RST\cong\triangle ABC\) (by matching \(R\to B\), \(S\to A\), \(T\to C\))

Answer:

\(\triangle RST\cong\triangle ABC\)