Question

1. choose the correct answer.

yw, wx, and xy are midsegments of △tuv
ty = 2.5 cm
ux = 3.7 cm
what is the measure of xv ?
○ 5 cm
○ 3.7 cm
○ 7.4 cm
○ 2.5 cm

Explanation:

Step1: Recall Midsegment Theorem

In a triangle, a midsegment connects the midpoints of two sides. So, \( X \) should be the midpoint of \( UV \) (since \( WX \) is a midsegment, or from the definition of midsegments in \( \triangle TUV \)). This means \( UX = XV \) because \( X \) is the midpoint, so it divides \( UV \) into two equal parts.

Step2: Use Given Length \( UX = 3.7 \, \text{cm} \)

Since \( X \) is the midpoint, \( XV = UX \). We know \( UX = 3.7 \, \text{cm} \), so \( XV = 3.7 \, \text{cm} \)? Wait, no, wait—wait, maybe I mixed up? Wait, no, let's check again. Wait, the midsegment: in \( \triangle TUV \), midsegments are \( YW \), \( WX \), \( XY \). So \( X \) is the midpoint of \( UV \), so \( UX = XV \). Wait, but the options: wait, maybe I made a mistake. Wait, no, the given \( TY = 2.5 \, \text{cm} \), \( UX = 3.7 \, \text{cm} \). Wait, maybe \( X \) is the midpoint of \( TV \)? Wait, no, the diagram: let's re-express. Wait, the triangle is \( TUV \), with midsegments \( YW \), \( WX \), \( XY \). So midsegments connect midpoints. So \( Y \) is midpoint of \( TV \), \( W \) is midpoint of \( TU \), \( X \) is midpoint of \( UV \). Wait, no, maybe \( X \) is midpoint of \( TV \)? Wait, no, the problem says "What is the measure of \( XV \)?". Let's see: if \( TY = 2.5 \, \text{cm} \), and \( Y \) is the midpoint of \( TV \), then \( TY = YV = 2.5 \, \text{cm} \), but that's not matching. Wait, maybe I messed up the midpoints. Wait, the other given is \( UX = 3.7 \, \text{cm} \). If \( X \) is the midpoint of \( UV \), then \( UX = XV \), so \( XV = 3.7 \, \text{cm} \)? But the options have 3.7 cm as an option. Wait, but let's check again. Wait, maybe the midsegment: in a triangle, midsegment is parallel to the third side and half its length. But here, we need to find \( XV \). Wait, maybe \( X \) is the midpoint of \( TV \), so \( TX = XV \). Wait, but \( TY = 2.5 \, \text{cm} \), no. Wait, the diagram: \( T \), \( U \), \( V \), \( Y \), \( W \), \( X \). Let's assume that \( X \) is the midpoint of \( TV \), so \( TX = XV \). But \( TY = 2.5 \, \text{cm} \), no. Wait, maybe the key is that \( X \) is the midpoint of \( UV \), so \( UX = XV \), so \( XV = 3.7 \, \text{cm} \). Wait, but the options: 3.7 cm is an option. Wait, but let's confirm. The midsegment theorem: midpoints divide the sides into two equal parts. So if \( UX = 3.7 \, \text{cm} \), and \( X \) is the midpoint of \( UV \), then \( XV = UX = 3.7 \, \text{cm} \). Wait, but maybe I made a mistake. Wait, no, the answer should be 3.7 cm? Wait, no, wait, maybe \( X \) is the midpoint of \( TV \), so \( TX = XV \), but \( TY = 2.5 \, \text{cm} \), no. Wait, the given \( TY = 2.5 \, \text{cm} \), \( UX = 3.7 \, \text{cm} \). Wait, maybe the correct answer is 2.5 cm? No, that doesn't make sense. Wait, let's re-express:

Wait, the problem says \( YW \), \( WX \), \( XY \) are midsegments of \( \triangle TUV \). So midsegments connect midpoints. So:

- \( Y \) is the midpoint of \( TV \) (since \( YW \) is a midsegment, so \( Y \) is midpoint of \( TV \), \( W \) is midpoint of \( TU \))
- \( X \) is the midpoint of \( UV \) (since \( WX \) is a midsegment, so \( W \) is midpoint of \( TU \), \( X \) is midpoint of \( UV \))
- \( Y \) is midpoint of \( TV \), \( X \) is midpoint of \( UV \), so \( XY \) is midsegment.

Wait, but we need to find \( XV \). If \( X \) is the midpoint of \( UV \), then \( UX = XV \). Given \( UX = 3.7 \, \text{cm} \), so \( XV = 3.7 \, \text{cm} \). Wait, but the options have 3.7 cm as an option. Wait, but let's check the other options. Wait, maybe I messed up the midpoint. Wait, maybe \( X \) is the midpoint of \( TV \), so \( TX = XV \), but \( TY = 2.5 \, \text{cm} \), no. Wait, the diagram shows \( T \) to \( V \) as 5 cm? Wait, the diagram has \( T \) to \( V \) as 5 cm? Wait, no, the user's diagram: "T 5 cm V"—wait, maybe \( TV = 5 \, \text{cm} \), but no. Wait, maybe the correct answer is 2.5 cm? No, that's \( TY \). Wait, no, let's start over.

Midsegment Theorem: A midsegment of a triangle is parallel to the third side and half as long as the third side. Also, the midpoint divides the side into two equal segments.

In \( \triangle TUV \), \( YW \), \( WX \), \( XY \) are midsegments. So:

- \( W \) is the midpoint of \( TU \), so \( TW = WU \)
- \( X \) is the midpoint of \( UV \), so \( UX = XV \)
- \( Y \) is the midpoint of \( TV \), so \( TY = YV \)

Given \( UX = 3.7 \, \text{cm} \), since \( X \) is the midpoint of \( UV \), \( XV = UX = 3.7 \, \text{cm} \).

Wait, but the options: 3.7 cm is an option (the second option: 3.7 cm). So that's the answer.

Answer:

\(\boldsymbol{3.7 \, \text{cm}}\) (the option with "3.7 cm")