Proportionality Theorems

Unit 7: Similarity · ~15–30 min

The Triangle Proportionality Theorem

Draw a line parallel to one side of a triangle so that it crosses the other two sides. That single parallel line splits both of those sides in the same ratio. This is the Triangle Proportionality Theorem, often called the side-splitter theorem.

• Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, it divides those two sides proportionally.
• Converse: If a line divides two sides of a triangle proportionally, it is parallel to the third side.

The parallel line also creates a smaller triangle similar to the original by AA Similarity — the corresponding angles are congruent because the parallel line creates equal corresponding angles with each transversal side.

Midsegments, revisited

Recall from Unit 5 that a midsegment connects the midpoints of two sides of a triangle, and is parallel to the third side with half its length. That's actually a special case of the Triangle Proportionality Theorem, where the "ratio" the parallel line creates is exactly 1:1 on both sides — each side is split into two congruent halves.

Guided practice

• 1.In ▵PQR, segment ST ∥ QR with PS = 5, SQ = 15, PT = 4. Find TR.
  Show answer

  5/15 = 4/TR, so 5(TR) = 60, TR = 12.
• 2.In ▵XYZ, a segment parallel to YZ splits XY into pieces of length 8 and 12 (from X). If the corresponding piece of XZ nearest X is 10, find the remaining piece of XZ.
  Show answer

  8/12 = 10/x, so 8x = 120, x = 15.
• 3.D and E are midpoints of sides AB and AC of ▵ABC. If DE = 7.5, find BC.
  Show answer

  DE is half of BC, so BC = 2(7.5) = 15.
• 4.In ▵LMN, a line crosses side LM at a point splitting it 3:5 (from L) and crosses LN at a point splitting it 6:x (from L). Find x if the crossing line is parallel to MN.
  Show answer

  3/5 = 6/x, so 3x = 30, x = 10.
• 5.Explain, using similar triangles, why a midsegment is always exactly half the length of the third side (not just parallel to it).
  Show answer

  Connecting the midpoints splits both sides in a 1:1 ratio, which means the small triangle cut off is similar to the original by SAS∼ with a scale factor of 1:2 (since each half-side is half the full side). Because the triangles are similar with that scale factor, every side of the small triangle — including the midsegment — is half the length of its corresponding side in the original, which is the third side.