Question

use the picture to name four rays.
o a) $overrightarrow{st}$, $overrightarrow{sw}$, $overrightarrow{su}$, $overrightarrow{uv}$
o b) $overrightarrow{vt}$, $overrightarrow{vw}$, $overrightarrow{wu}$, $overrightarrow{vs}$
o c) $overrightarrow{st}$, $overrightarrow{tu}$, $overrightarrow{tv}$, $overrightarrow{tw}$
o d) $overrightarrow{ws}$, $overrightarrow{wt}$, $overrightarrow{wu}$, $overrightarrow{vw}$

Explanation:

To name a ray, we use the notation \(\overrightarrow{AB}\) where \(A\) is the endpoint and \(B\) is a point on the ray (indicating the direction). Let's analyze each option:

Step 1: Analyze Option A

• \(\overrightarrow{ST}\): Endpoint \(S\), direction towards \(T\) – valid ray.
• \(\overrightarrow{SW}\): Endpoint \(S\), direction towards \(W\) – valid ray.
• \(\overrightarrow{SU}\): Endpoint \(S\), direction towards \(U\) – valid ray.
• \(\overrightarrow{UV}\): Endpoint \(U\), direction towards \(V\) – but \(V\) is not on the same line extending from \(U\) as per the diagram (the horizontal line has \(S, T, U\) and the other lines are from \(T\) to \(V\) and \(T\) to \(W\)). So this is invalid.

Step 2: Analyze Option B

• \(\overrightarrow{VT}\): Endpoint \(V\), direction towards \(T\) – valid ray (from \(V\) to \(T\)).
• \(\overrightarrow{VW}\): Endpoint \(V\), direction towards \(W\) – but \(V\) and \(W\) are not on a straight line (from diagram, \(V\) is on the left line from \(T\), \(W\) on the right line from \(T\)). Invalid.
• \(\overrightarrow{WU}\): Endpoint \(W\), direction towards \(U\) – valid ray (from \(W\) to \(U\) along the horizontal? Wait, no, \(W\) is on the right line from \(T\), \(U\) is on the horizontal. Maybe not. But let's check other options.
• \(\overrightarrow{VS}\): Endpoint \(V\), direction towards \(S\) – but \(V\) to \(S\) is not a ray as per diagram. Invalid.

Step 3: Analyze Option C

• \(\overrightarrow{ST}\): Endpoint \(S\), direction \(T\) – valid.
• \(\overrightarrow{TU}\): Endpoint \(T\), direction \(U\) – valid (horizontal line).
• \(\overrightarrow{TV}\): Endpoint \(T\), direction \(V\) – valid (left line from \(T\) to \(V\)).
• \(\overrightarrow{TW}\): Endpoint \(T\), direction \(W\) – valid (right line from \(T\) to \(W\)). All these rays have \(T\) or \(S\) as endpoints and the other points on the ray. This seems correct.

Step 4: Analyze Option D

• \(\overrightarrow{WS}\): Endpoint \(W\), direction \(S\) – \(W\) to \(S\) is not a ray (diagram shows \(W\) on right line from \(T\), \(S\) on left end of horizontal). Invalid.
• \(\overrightarrow{WT}\): Endpoint \(W\), direction \(T\) – valid (from \(W\) to \(T\)).
• \(\overrightarrow{WU}\): Endpoint \(W\), direction \(U\) – valid (from \(W\) to \(U\) horizontal).
• \(\overrightarrow{VW}\): Endpoint \(V\), direction \(W\) – \(V\) and \(W\) not on same line. Invalid.

Answer:

C) \(\overrightarrow{ST}\), \(\overrightarrow{TU}\), \(\overrightarrow{TV}\), \(\overrightarrow{TW}\)