Question

1. match the directed line segment to the image of polygon p being transformed to polygon q by translation by that directed line segment.

Explanation:

Step1: Recall Translation Properties

A translation moves a figure by a directed line segment (vector), so corresponding vertices of \( P \) and \( Q \) should have the same displacement as the directed line segment.

Step2: Analyze Each Pair

- **Top Pair (First \( P \) to \( Q \))**: Observe the horizontal/vertical shift. The directed line segment \( a \) (horizontal, right) matches the shift from \( P \) to \( Q \) here? Wait, no—wait, first \( P \) (top left) to \( Q \) (right below? Wait, no, first \( P \) is top left triangle, \( Q \) is to its right and down? Wait, no, let's check coordinates (visually). The first \( P \) to \( Q \): the vector (directed line segment) should be from \( P \)'s vertex to \( Q \)'s corresponding vertex. Let's take the bottom vertex of \( P \): moving to bottom vertex of \( Q \)—horizontal right, vertical down? Wait, no, the directed line segments on the right: \( a \) is horizontal right, \( u \) is vertical up, \( v \) is diagonal (left-down? No, \( v \) is left to right? Wait, no, the arrows: \( a \) has arrow to right, \( u \) arrow up, \( v \) arrow to right-down? Wait, maybe better to match the direction and length.

- **Second Pair (Middle \( P \) to \( Q \))**: \( P \) is right, \( Q \) is left. So the vector should be leftward. Among the directed segments, \( v \) or \( w \)? Wait, \( v \) and \( w \) are diagonal? Wait, maybe I missee. Wait, the problem is to match each \( P \to Q \) translation with the correct directed line segment.

Let's list the \( P \) and \( Q \) pairs:

1. Top \( P \) (triangle) to top \( Q \) (triangle to its right/down? Wait, no, first row: \( P \) top left, \( Q \) to its right, same shape. The directed segment \( a \) is horizontal right. So this \( P \to Q \) is horizontal translation right, so matches \( a \).

2. Middle \( P \) (right) to middle \( Q \) (left). So translation left. The directed segment \( v \) (or \( w \))? Wait, \( v \) and \( w \) are diagonal? Wait, maybe the second \( P \) (middle right) to \( Q \) (middle left): the vector is leftward, so directed segment \( v \) (if \( v \) is left? Wait, no, the arrows: \( v \) has arrow to the right-down? Wait, maybe the third pair: \( P \) above \( Q \), so vertical translation down? No, \( u \) is vertical up. Wait, maybe I need to look at the direction of the directed line segment (from start to end) and the translation vector (from \( P \)'s vertex to \( Q \)'s corresponding vertex).

Let's correct:

- **First Translation (Top \( P \) to \( Q \))**: The displacement from \( P \)'s vertices to \( Q \)'s is horizontal right (same as directed segment \( a \), which is horizontal right). So match \( P \to Q \) (top) with \( a \).

- **Second Translation (Middle \( P \) (right) to \( Q \) (left))**: Displacement is horizontal left? Wait, no, the directed segments: \( v \) and \( w \) are diagonal. Wait, maybe the second \( P \) is middle right, \( Q \) is middle left: the vector is left and same vertical? Wait, no, the directed segment \( v \) (if \( v \) is leftward diagonal? Wait, the problem's directed segments: \( a \) (horizontal right), \( u \) (vertical up), \( v \) (diagonal left-down? No, arrow direction: \( a \) start at left, end at right; \( u \) start at bottom, end at top; \( v \) start at right, end at left-down? No, maybe \( v \) is start at left, end at right-down? Wait, the key is that translation vector is the same as the directed line segment (from \( P \) to \( Q \), the vector is \( Q - P \) for each vertex).

- **Third Translation (Lower \( P \) above \( Q \))**: \( P \) is above \( Q \), so vertical translation down? No, \( u \) is up. Wait, maybe the fourth pair: \( P \) bottom right, \( Q \) bottom left: displacement is left and down? No, this is getting confusing. But the standard way is: for each \( P \) and \( Q \), draw the vector from a vertex of \( P \) to the corresponding vertex of \( Q \), then match with the directed line segment (same direction and length).

Assuming the top \( P \to Q \) is horizontal right (matches \( a \)), middle \( P \) (right) to \( Q \) (left) is horizontal left? No, directed segments: \( a \) (right), \( u \) (up), \( v \) (diagonal left-down), \( w \) (diagonal right-down)? Wait, maybe the correct matches are:

1. Top \( P \to Q \): vector is horizontal right (matches \( a \)).

2. Middle \( P \) (right) \(\to Q\) (left): vector is horizontal left? No, directed segment \( v \) (if \( v \) is leftward). Wait, maybe the answer is:

- First \( P \to Q \): \( a \)

- Second \( P \to Q \): \( v \) (or \( w \))—but likely, the key is to identify the translation vector (direction and length) matching the directed line segment.

Answer:

(Assuming the top \( P \) to \( Q \) uses directed segment \( a \), middle \( P \) to \( Q \) uses \( v \), etc.—but since the problem is to match, the correct matches are based on translation vector direction/length. However, without exact coordinates, the standard approach is:

• The top \( P \) to \( Q \): translation right (matches \( a \)).

• The middle \( P \) (right) to \( Q \) (left): translation left (matches \( v \) if \( v \) is left).

• The lower \( P \) above \( Q \): translation down (but \( u \) is up—maybe vertical down? No, \( u \) is up. Wait, maybe the fourth \( P \) (bottom right) to \( Q \) (bottom left): translation left and down (matches \( w \)).

But since the problem is to match, the final answer would be the correct directed line segment for each \( P \to Q \), but as a general solution, the key is to use the translation vector (from \( P \) to \( Q \)) matching the directed line segment.

(Note: Due to the image's visual nature, the precise match requires identifying the displacement vector. For example, if \( P \) moves right to \( Q \), the directed segment is \( a \); if left, \( v \); vertical, \( u \); diagonal, \( w \).)