Question

which sequence of transformations will carry △ abc onto △ abc?
a rotation 180 degrees and translation 1 unit left
b translation left 3 units and translation up 1 unit
c reflection over the x axis
d translation left 1 units and translation down 3 units

Explanation:

Step1: Analyze each option

• Option A: Rotation 180 degrees changes the orientation of the triangle in a way that is not consistent with the given transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).
• Option B: Translation left 3 units and up 1 unit. Let's assume a general point \((x,y)\) in \(\triangle ABC\). After translation left 3 units \((x - 3)\) and up 1 unit \((y+1)\). If we take a vertex of \(\triangle ABC\) say \(A(-1,4)\), after translation left 3 units \(x=-1-3=-4\) and up 1 unit \(y = 4 + 1=5\) which is not the position of the corresponding vertex in \(\triangle A'B'C'\).
• Option C: Reflection over the \(x\) - axis changes the \(y\) - coordinate sign \((x,y)\to(x, - y)\). For example, if \(A(-1,4)\) then after reflection over \(x\) - axis it would be \((-1,-4)\) which is not correct.
• Option D: Translation left 1 unit \((x-1)\) and down 3 units \((y - 3)\). Take vertex \(A(-1,4)\): \(x=-1-1=-2\), \(y=4 - 3 = 1\). Take vertex \(B(1,3)\): \(x=1-1 = 0\), \(y=3-3=0\). Take vertex \(C(-2,1)\): \(x=-2-1=-3\), \(y=1 - 3=-2\).

Answer:

D. translation left 1 unit and translation down 3 units