Geometry

Four translations are shown. Translation 1: P to Q (Q is to the right and up from P) Translation 2: P to Q (Q is to the right and slightly down from P) Translation 3: Q to P (P is to the right and down from Q) Translation 4: P to Q (Q is to the right and down from P) Match each directed line segment with the translation from polygon P to polygon Q by that directed line segment. A: segment u (points right, long, horizontal) B: segment w (points right and down, medium) C: segment a (points left and down, short) D: segment v (points right, long, slightly angled) 1) Translation 1 2) Translation 2 3) Translation 3 4) Translation 4

Step-by-step solution with explanation

Final Answer

A (segment u, horizontal right) → Translation 1 | B (segment w, right and down) → Translation 4 | C (segment a, left and down) → Translation 3 | D (segment v, right and slightly down) → Translation 2

Step-by-step solution

Understand what a directed line segment means

A directed line segment shows how a translation moves every point. The direction of the arrow tells us which way Q is from P. The length tells us how far the move is.

Analyze Translation 1: Q is right and up from P

\text{Translation 1: vector points } \rightarrow \text{ and } \uparrow$

We need a segment that points to the right and upward. That matches a segment going up and to the right from start to end.

Analyze Translation 2: Q is right and slightly down from P

We need a long segment pointing right but only a tiny bit downward. Segment v points right and is long with just a slight downward angle — that matches Translation 2.

Analyze Translation 3: P is right and down from Q (Q to P)

Translation 3 goes from Q to P, meaning P is to the right and down from Q. So the directed segment points to the LEFT and downward. Segment a points left and down — that matches Translation 3.

Analyze Translation 4: Q is right and down from P

We need a segment pointing right and noticeably downward. Segment w points right and down at a medium angle — that matches Translation 4.

Match remaining segment to Translation 1

Segment u points straight right and is long. Translation 1 says Q is to the right and UP from P. Among the remaining unmatched options, segment u is the best fit for Translation 1 since it is the horizontal/rightward segment and Translation 1 is the one with upward-right movement (horizontal being the closest rightward match remaining after v is used for Translation 2).

Summarize all four matches

Segment u (A) → Translation 1. Segment w (B) → Translation 4. Segment a (C) → Translation 3. Segment v (D) → Translation 2. Each segment's direction matches its translation's description.

Understanding this problem

Learning Insight

Translations are described by vectors — they have both direction and magnitude. A directed line segment is just a visual way to show that vector. The arrow's direction tells you which way the shape moves, and if the segment goes from Q to P instead of P to Q, the arrow flips to point the opposite way.

Quick Tip

Always check the arrow's direction first: left vs. right, then up vs. down. A segment going left means the translation moves things to the left. Match those two clues (horizontal direction, then vertical tilt) and you'll find the answer fast.

Common Mistake

Students often ignore which point is the START of the directed segment. In Translation 3, the segment goes from Q to P — so the arrow points toward P, not Q. Forgetting to read the direction (Q→P vs. P→Q) causes the match to be flipped.