Question

which statement proves that the triangles with the given vertices are congruent. a(3,1), b(4,5), c(2,3) and a(-1,-3), b(-5,-4), c(-3,-2) a the triangles are congruent because △abc can be mapped onto △abc by a translation: (x,y)→(x - 4,y - 6). b the triangles are congruent because △abc can be mapped onto △abc by a rotation: (x,y)→(y, - x), followed by a reflection. c the triangles are congruent because △abc can be mapped onto △abc by a reflection: (x,y)→(-x,y), followed by a rotation. d the triangles are congruent because △abc can be mapped onto △abc by a rotation: (x,y)→(-y,x), followed by a reflection.

Explanation:

Step1: Check translation formula for each point

For translation \((x,y)\to(x - 4,y - 6)\):

• For point \(A(3,1)\): \(x=3,y = 1\). After translation \(x'=3-4=-1,y'=1 - 6=-3\), so \(A'\) is \((-1,-3)\)
• For point \(B(4,5)\): \(x = 4,y=5\). After translation \(x'=4-4 = 0

eq-5\) (This is wrong for option A's translation claim)

Step2: Check reflection formula

For reflection \((x,y)\to(-x,y)\) followed by rotation (not relevant as we first check individual transformations). For example, if we consider option C's reflection \((x,y)\to(-x,y)\) on \(A(3,1)\) gives \((-3,1)
eq A'(-1,-3)\) (Wrong)

Step3: Check rotation - reflection combination

For rotation \((x,y)\to(-y,x)\) followed by reflection (for option B). For \(A(3,1)\), rotation gives \((-1,3)\), then further transformation (not matching \(A'(-1,-3)\) easily. Let's check option D's rotation - reflection
For rotation \((x,y)\to(-y,x)\) (first part of option D's transformation). For \(A(3,1)\), we get \((-1,3)\). Then reflection over \(x\) - axis \((x,y)\to(x,-y)\) gives \((-1,-3)\) (matches \(A'\)). For \(B(4,5)\): Rotation \((x,y)\to(-y,x)\) gives \((-5,4)\), reflection over \(x\) - axis \((x,y)\to(x,-y)\) gives \((-5,-4)\) (matches \(B'\)). For \(C(2,3)\): Rotation \((x,y)\to(-y,x)\) gives \((-3,2)\), reflection over \(x\) - axis \((x,y)\to(x,-y)\) gives \((-3,-2)\) (matches \(C'\))

Answer:

D. The triangles are congruent because \(\triangle ABC\) can be mapped onto \(\triangle A'B'C'\) by a rotation \((x,y)\to(-y,x)\) followed by a reflection \((x,y)\to(x, - y)\)