Problem preview

G-SRT.4 Warmup M2-049-A03-V04

Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.

Use the converse of the side-splitter theorem to decide whether a segment is parallel to the third side of a triangle

Problem

In triangle \(XYZ\), point \(M\) is on \(XY\) and point \(N\) is on \(XZ\). Use the converse of the side-splitter theorem to decide whether segment \(MN\) must be parallel to \(YZ\) if \(XM/MY~=~7/14\) and \(XN/NZ~=~3/6\).

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Yes, because the ratios are equal, but that only shows proportional sides and does not prove that MN is parallel to YZ.

No, because you need to compare XM/XY and XN/XZ, not XM/MY and XN/NZ, so MN is not guaranteed to be parallel to YZ.

No, because 7/14 and 3/6 use different numbers, so there is not enough information to say MN is parallel to YZ.

Yes, because 7/14 and 3/6 both simplify to 1/2, the sides are divided proportionally, so MN is parallel to YZ.

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Curriculum context

Course: Math II
Standard: G-SRT.4
Category: Geometry
Domain: Similarity, Right Triangles, and Trigonometry
Objective: Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Problem type: Use the converse of the side-splitter theorem to decide whether a segment is parallel to the third side of a triangle