Question

what is the missing step in the sequence of transformations that will map figure 1 onto figure 2?
1.?

1. translation 3 units to the right
2. translation 4 units down

graph of coordinate plane with two triangles: triangle 1 (orange) at left, triangle 2 (pink) at right. axes labeled, grid present.
a. rotation 270° clockwise about the origin
b. reflection across the origin
c. rotation 90° clockwise about the origin
d. reflection across the x - axis

Explanation:

To determine the missing transformation, we analyze the orientation of Figure 1 and Figure 2. Figure 1 is an upward - facing triangle on the left, and Figure 2 is a triangle on the right with a different orientation.

• Option A: A \(270^{\circ}\) clockwise rotation about the origin is equivalent to a \(90^{\circ}\) counter - clockwise rotation. This would not give the correct orientation for Figure 2.
• Option B: A reflection across the origin would map a point \((x,y)\) to \((-x,-y)\). This does not match the transformation needed to get from Figure 1 to Figure 2.
• Option C: A \(90^{\circ}\) clockwise rotation about the origin. Let's consider a vertex of Figure 1, say the left - most vertex at \((-6,0)\). After a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\), \((-6,0)\) becomes \((0,6)\). Then, after translation 3 units right (\(x\) - coordinate increases by 3) and 4 units down (\(y\) - coordinate decreases by 4), we get \((0 + 3,6-4)=(3,2)\), which is close to the vertices of Figure 2. Another vertex, the top vertex of Figure 1 at \((-4,4)\). After \(90^{\circ}\) clockwise rotation: \((4,4)\). After translation 3 right and 4 down: \((4 + 3,4 - 4)=(7,0)\), which is also consistent with Figure 2's vertices.
• Option D: A reflection across the \(x\) - axis would map \((x,y)\) to \((x,-y)\). For the top vertex \((-4,4)\) of Figure 1, it would become \((-4,-4)\), which is not consistent with Figure 2's vertices.

So the missing step is a rotation \(90^{\circ}\) clockwise about the origin.

Answer:

C. rotation \(90^{\circ}\) clockwise about the origin