Answered

Points S, U, and T are the midpoints of the sides of △PQR.



Which statements are correct? Check all that apply.



□ $\frac{1}{2} QP = UT$

□ $\frac{1}{2} TS = RQ$

□ $SU = PR$

□ $\overline{ SU } // \overline{ RP }$

□ $\overline{U T} \perp \overline{R P}$

Verified Answer :

  • Use the triangle midsegment theorem to establish that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
  • Identify the relations: \(UT = \frac{1}{2} PQ\), \(TS = \frac{1}{2} QR\), and \(SU = \frac{1}{2} PR\), with corresponding parallel lines.
  • Evaluate the statements: \(\frac{1}{2} QP = UT\) and \(\overline{SU} \parallel \overline{RP}\) are consistent with the theorem.
  • The correct statements are: \(\boxed{\frac{1}{2} QP = UT \text{ and } \overline{SU} \parallel \overline{RP}}\).

Explanation

  1. Apply the Triangle Midsegment Theorem

In \(\triangle PQR\), let \(S\), \(U\), and \(T\) be the midpoints of sides \(PQ\), \(QR\), and \(PR\), respectively. The triangle midsegment theorem states that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is exactly half the length of that third side.

Based on the definitions provided:

  • Segment \(UT\) connects the midpoints of \(QR\) and \(PR\). Therefore, it is parallel to the third side \(PQ\) and its length is half the length of \(PQ\). That is, \(UT = \frac{1}{2} PQ\).
  • Segment \(TS\) connects the midpoints of \(PR\) and \(PQ\). Therefore, it is parallel to the third side \(QR\) and its length is half the length of \(QR\). That is, \(TS = \frac{1}{2} QR\).
  • Segment \(SU\) connects the midpoints of \(PQ\) and \(QR\). Therefore, it is parallel to the third side \(PR\) and its length is half the length of \(PR\). That is, \(SU = \frac{1}{2} PR\).
  1. Evaluate the given statements

Now, let's evaluate each of the given statements based on these geometric relationships:

  1. \(\frac{1}{2} QP = UT\): This matches our derived property \(UT = \frac{1}{2} PQ\) (or \(QP\)). This statement is correct.
  2. \(\frac{1}{2} TS = RQ\): From our derivation, \(TS = \frac{1}{2} RQ\) (or \(QR\)). The statement \(\frac{1}{2} TS = RQ\) implies \(TS = 2 RQ\), which contradicts the theorem. This statement is incorrect.
  3. \(SU = PR\): From our derivation, \(SU = \frac{1}{2} PR\). This statement is incorrect.
  4. \(\overline{SU} \parallel \overline{RP}\): Our derivation showed that \(SU\) is parallel to \(PR\) (or \(RP\)). This statement is correct.
  5. \(\overline{UT} \perp \overline{RP}\): The midsegment theorem only guarantees parallelism; it does not impose a perpendicularity condition unless specific properties of the original triangle (like being a right triangle) are given. There is no such information here. This statement is incorrect.
  1. Conclusion of correct statements

Based on the evaluation, the correct statements are:

  • \(\frac{1}{2} QP = UT\)
  • \(\overline{SU} \parallel \overline{RP}\)

Thus, the final identified correct statements are \(\boxed{\frac{1}{2} QP = UT \text{ and } \overline{SU} \parallel \overline{RP}}\).

Examples

The triangle midsegment theorem is highly useful in fields like civil engineering and structural design. For instance, if you are designing a support frame for a roof in a triangular shape and need to add a cross-brace that connects the midpoints of two sides, the theorem allows you to immediately determine that the brace will be parallel to the roof's base and exactly half its length. This simple relationship helps engineers ensure stability and symmetry in their designs without needing to perform complex measurements for every individual component.