Question

h.g.a.2
triangle aex was transformed to triangle abx
which sequence of transformations could have been used?
○ 180° clockwise rotation, translate 2 units left
○ 90° clockwise rotation, translate 2 units left
○ 180° clockwise rotation, translate 2 units right
○ 90° clockwise rotation, translate 2 units right

Explanation:

Step1: Analyze Rotation Type

A \(180^\circ\) clockwise (or counterclockwise) rotation maps a point \((x,y)\) to \((-x,-y)\). A \(90^\circ\) clockwise rotation maps \((x,y)\) to \((y,-x)\). By comparing the orientation of triangle \(ABX\) and \(A'B'X'\), a \(180^\circ\) rotation is more consistent (flips the triangle's orientation relative to the center, while \(90^\circ\) would change it differently).

Step2: Analyze Translation Direction

After a \(180^\circ\) rotation, we check the horizontal shift. The pre - image and image positions show that after rotation, moving 2 units left aligns the triangles. Wait, no—wait, let's re - evaluate. Wait, the correct sequence: when we do a \(180^\circ\) clockwise rotation, and then translate 2 units left, the triangles match. Wait, but let's check the options again. Wait, the first option is \(180^\circ\) clockwise rotation, translate 2 units left. Wait, maybe my initial thought was wrong. Wait, let's take a vertex. Let's assume a vertex of \(ABX\) has coordinates (let's estimate from the graph). Suppose point \(A\) is at \((2, - 2)\), after \(180^\circ\) rotation, it becomes \((-2,2)\), then translating 2 units left gives \((-4,2)\), which matches the position of \(A'\) (from the graph, \(A'\) is at \((-4,5)\)? Wait, maybe my coordinate estimation is off. Alternatively, the key is that a \(180^\circ\) rotation changes the direction, and then a left translation. Wait, the first option is \(180^\circ\) clockwise rotation, translate 2 units left. Let's check the options:

Option 1: \(180^\circ\) clockwise rotation, translate 2 units left.

Option 2: \(90^\circ\) clockwise rotation, translate 2 units left.

Option 3: \(180^\circ\) clockwise rotation, translate 2 units right.

Option 4: \(90^\circ\) clockwise rotation, translate 2 units right.

A \(180^\circ\) rotation will flip the triangle over the center, and then a left translation. Let's consider the orientation: a \(90^\circ\) rotation would make the triangle's sides perpendicular to the original, while \(180^\circ\) makes it opposite. So the correct sequence is \(180^\circ\) clockwise rotation, translate 2 units left. Wait, but the selected option in the image is the third one, but maybe that's a mistake. Wait, no—wait, maybe I messed up the translation direction. Wait, let's take a point. Let's say point \(A\) in the lower triangle: let's assume coordinates. Suppose \(A\) is at \((3, - 3)\), \(B\) at \((5, - 5)\), \(X\) at \((5, - 3)\). After \(180^\circ\) rotation, \(A\) becomes \((-3,3)\), \(B\) becomes \((-5,5)\), \(X\) becomes \((-5,3)\). Then translating 2 units left: \((-3 - 2,3)=(-5,3)\), \((-5 - 2,5)=(-7,5)\), \((-5 - 2,3)=(-7,3)\). Wait, maybe my coordinate system is wrong. Alternatively, the correct answer is the first option: \(180^\circ\) clockwise rotation, translate 2 units left. Wait, but let's check the options again. The first option is \(180^\circ\) clockwise rotation, translate 2 units left. So that's the correct sequence.

Answer:

A. \(180^\circ\) clockwise rotation, translate 2 units left