Question

which transformation or sequence of transformations maps △abc to △abc? a(5,5) b(1,3) c(2,1) c(-3,-1) b(-4,-3) a(0,-5) a a translation 5 units left b a reflection across x = 0, then a translation 5 units left c a reflection across y = 0, then a translation 5 units left d a rotation 90° about the origin

Explanation:

Step1: Analyze translation of point A

For point A(5, 5) to A'(0, - 5), in the x - direction, the change is \(0 - 5=-5\) (5 units left), and in the y - direction, the change is \(-5 - 5=-10\). A translation 5 units left only affects the x - coordinate and not the y - coordinate as seen here, so option A is incorrect.

Step2: Analyze reflection and translation in option B

A reflection across \(x = 0\) (the y - axis) changes the sign of the x - coordinate. For point A(5, 5), after reflection across \(x = 0\), it becomes (-5, 5). Then a translation 5 units left makes it \((-5-5,5)=(-10,5)
eq A'(0,-5)\), so option B is incorrect.

Step3: Analyze reflection and translation in option C

A reflection across \(y = 0\) (the x - axis) changes the sign of the y - coordinate. For point A(5, 5), after reflection across \(y = 0\), it becomes (5, - 5). Then a translation 5 units left makes it \((5 - 5,-5)=(0,-5)\). For point B(1, 3), after reflection across \(y = 0\) it is (1,-3), and then 5 units left gives \((1 - 5,-3)=(-4,-3)\). For point C(2, 1), after reflection across \(y = 0\) it is (2,-1), and then 5 units left gives \((2 - 5,-1)=(-3,-1)\). This sequence of transformations maps \(\triangle ABC\) to \(\triangle A'B'C'\).

Step4: Analyze rotation in option D

A 90 - degree rotation about the origin has the transformation rule \((x,y)\to(-y,x)\). For point A(5, 5), it would become (-5, 5)\(
eq A'(0,-5)\), so option D is incorrect.

Answer:

C. a reflection across \(y = 0\), then a translation 5 units left