Question

5 △def ≅ △lmn. which sequence of transformations can be performed on △def to show this?
image of coordinate grid with triangles lmn (top left) and def (bottom right)
a reflect △def across the y-axis. then translate the image 6 units up.
b translate △def 6 units up. then rotate the image 90° counterclockwise around the origin.
c rotate △def 90° clockwise around the origin. then rotate the image 90° clockwise around the origin.
d translate △def 6 units to the left. then rotate the image 90° clockwise around the origin.

Explanation:

Step1: Analyze Option A

First, reflect \(\triangle DEF\) across the \(y\)-axis. The coordinates of \(D(1, - 4)\), \(E(5, - 2)\), \(F(1, - 2)\) after reflection over \(y\)-axis become \(D'(-1, - 4)\), \(E'(-5, - 2)\), \(F'(-1, - 2)\). Then translate 6 units up: \(D''(-1, - 4 + 6)=(-1,2)\), \(E''(-5, - 2+6)=(-5,4)\), \(F''(-1, - 2 + 6)=(-1,4)\). Now check \(\triangle LMN\) coordinates: \(L(-1,4)\), \(M(-5,2)\), \(N(-1,2)\). Wait, let's re - check the reflection and translation. Wait, original \(\triangle DEF\): Let's get correct coordinates. From the graph, \(D(1,-4)\), \(F(1,-2)\), \(E(5,-2)\). Reflect over \(y\)-axis: \(D(-1,-4)\), \(F(-1,-2)\), \(E(-5,-2)\). Then translate 6 units up: \(D(-1,-4 + 6)=(-1,2)\), \(F(-1,-2 + 6)=(-1,4)\), \(E(-5,-2 + 6)=(-5,4)\). Now \(\triangle LMN\): \(L(-1,4)\), \(M(-5,2)\), \(N(-1,2)\). Wait, \(L\) is \((-1,4)\), \(N\) is \((-1,2)\), \(M\) is \((-5,2)\). So after reflection and translation, \(D(-1,2)\), \(F(-1,4)\), \(E(-5,4)\) – wait, \(L\) is \((-1,4)\), \(N\) is \((-1,2)\), \(M\) is \((-5,2)\). So if we consider the triangle, after reflection over \(y\)-axis and translation 6 units up, the triangle matches \(\triangle LMN\) (since \(D\) moves to \(N\)'s \(x\) and \(y\) adjusted, \(F\) moves to \(L\), \(E\) moves to \(M\)? Wait, maybe I mixed up the vertices. But let's check other options.

Step2: Analyze Option B

Translate \(\triangle DEF\) 6 units up: \(D(1,-4 + 6)=(1,2)\), \(F(1,-2 + 6)=(1,4)\), \(E(5,-2 + 6)=(5,4)\). Then rotate \(90^{\circ}\) counter - clockwise around origin. The rotation formula for a point \((x,y)\) is \((-y,x)\). So \(D(1,2)\) becomes \((-2,1)\), \(F(1,4)\) becomes \((-4,1)\), \(E(5,4)\) becomes \((-4,5)\). This does not match \(\triangle LMN\)'s coordinates (\(L(-1,4)\), \(M(-5,2)\), \(N(-1,2)\)).

Step3: Analyze Option C

Rotating \(90^{\circ}\) clockwise twice is equivalent to rotating \(180^{\circ}\) clockwise. The formula for \(90^{\circ}\) clockwise rotation is \((y, - x)\), and for \(180^{\circ}\) is \((-x,-y)\). For \(D(1,-4)\), after \(180^{\circ}\) rotation: \((-1,4)\), \(F(1,-2)\) becomes \((-1,2)\), \(E(5,-2)\) becomes \((-5,2)\). Wait, but the order of rotation: first \(90^{\circ}\) clockwise, then another \(90^{\circ}\) clockwise. First rotation (\(90^{\circ}\) clockwise) of \(D(1,-4)\): \((-4,-1)\), then another \(90^{\circ}\) clockwise: \((-1,4)\). \(F(1,-2)\): first \(90^{\circ}\) clockwise: \((-2,-1)\), then another \(90^{\circ}\) clockwise: \((-1,2)\). \(E(5,-2)\): first \(90^{\circ}\) clockwise: \((-2,-5)\), then another \(90^{\circ}\) clockwise: \((-5,2)\). But the triangle formed by these points is not the same as \(\triangle LMN\) in terms of vertex order? Wait, no, the problem is about congruence, but the transformation sequence. But let's check option A again.

Wait, going back to option A: Reflect \(\triangle DEF\) over \(y\)-axis: \(D(1,-4)\to(-1,-4)\), \(F(1,-2)\to(-1,-2)\), \(E(5,-2)\to(-5,-2)\). Then translate 6 units up: \((-1,-4 + 6)=(-1,2)\), \((-1,-2 + 6)=(-1,4)\), \((-5,-2 + 6)=(-5,4)\). Now \(\triangle LMN\) has \(L(-1,4)\), \(M(-5,2)\), \(N(-1,2)\). So the points after transformation: \(D''(-1,2)\) (which is \(N\)), \(F''(-1,4)\) (which is \(L\)), \(E''(-5,4)\) (which is \(M\))? Wait, maybe the vertices are mapped as \(D\to N\), \(F\to L\), \(E\to M\). Since the triangles are congruent, the transformation sequence in option A works.

Option D: Translate 6 units left: \(D(1 - 6,-4)=(-5,-4)\), \(F(1 - 6,-2)=(-5,-2)\), \(E(5 - 6,-2)=(-1,-2)\). Then rotate \(90^{\circ}\) clockwise: for \(D(-5,-4)\), rotation is \((-4,5)\); \(F(-5,-2)\) is \((-2,5)\); \(E(-1,-2)\) is \((-2,1)\). This does not match \(\triangle LMN\).

Answer:

A. Reflect \(\triangle DEF\) across the \(y\)-axis. Then translate the image 6 units up.