Question

1. given that δuvt is the mid - segment δ, tell which of the following are true. select all that apply. 2(tu)=pn mn || uv 2mp = tv ut || vn

Explanation:

Step1: Recall mid - segment theorem

In a triangle, a mid - segment (a line segment joining the mid - points of two sides of a triangle) is parallel to the third side and its length is half of the length of the third side.

Step2: Analyze \(2(TU)=PN\)

Since \(\triangle UVT\) is a mid - segment triangle, \(TU\) is a mid - segment parallel to \(PN\). By the mid - segment theorem, \(TU=\frac{1}{2}PN\), so \(2(TU) = PN\).

Step3: Analyze \(MN\parallel UV\)

\(UV\) is a mid - segment of a larger triangle formed by the points related to the figure. Mid - segments are parallel to the non - adjacent sides of the triangle they are part of. So \(MN\parallel UV\).

Step4: Analyze \(2MP = TV\)

There is no mid - segment relationship that would imply \(2MP=TV\). \(MP\) and \(TV\) are not related in the way described by the mid - segment theorem.

Step5: Analyze \(UT\parallel VN\)

\(UT\) is a mid - segment of a relevant triangle, and by the mid - segment theorem, \(UT\) is parallel to \(VN\).

Answer:

1. \(2(TU)=PN\): True
2. \(MN\parallel UV\): True
3. \(2MP = TV\): False
4. \(UT\parallel VN\): True