Question

△ abc and △ def are graphed on a coordinate plane. which sequence of transformations maps △ abc onto △ def to prove they are congruent? a. a reflection over the y-axis, and then a translation right one unit b. a translation right one unit, and then a reflection over the y-axis c. a rotation 90° clockwise about the origin, and then a translation right one unit d. a rotation 180° about the origin, and then a reflection over the x-axis

Explanation:

Step1: Analyze Reflection over y - axis

First, consider reflecting \(\triangle ABC\) over the \(y\) - axis. The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). Let's take the coordinates of the vertices of \(\triangle ABC\). Let's assume \(C=( - 1,0)\), \(B=( - 2,4)\), \(A=( - 4,2)\). After reflection over the \(y\) - axis, \(C\) becomes \((1,0)\), \(B\) becomes \((2,4)\), \(A\) becomes \((4,2)\).

Step2: Analyze Translation right 1 unit

Then, translate the reflected triangle right 1 unit. The rule for translation right 1 unit is \((x,y)\to(x + 1,y)\). So, the new coordinates: \(C=(1 + 1,0)=(2,0)\) (which is the coordinate of \(F\)), \(B=(2+1,4)=(3,4)\) (which is the coordinate of \(E\)), \(A=(4 + 1,2)=(5,2)\) (which is the coordinate of \(D\)). So, a reflection over the \(y\) - axis and then a translation right one unit maps \(\triangle ABC\) onto \(\triangle DEF\).

Let's check option B: If we translate right 1 unit first, then reflect over \(y\) - axis, the result will not match. For option C, a \(90^{\circ}\) clockwise rotation about the origin has the rule \((x,y)\to(y,-x)\), which will not map the triangle to the correct position. For option D, a \(180^{\circ}\) rotation about the origin has the rule \((x,y)\to(-x,-y)\), and then reflection over \(x\) - axis \((x,y)\to(x,-y)\) will also not give the correct mapping.

Answer:

A. a reflection over the \(y\) - axis, and then a translation right one unit