Mid-point Theorem and Its Converse [Including Intercept Theorem]

227. The Equal Intercept Theorem · Parallel lines cut transversals in matching ratios

Drag P and L to slant the two transversals any way you like. Because the parallel lines l, m, n are equally spaced, each transversal is cut into equal parts: PQ = QR and LM = MN.

lmnPQ = 152.6PQ = 152.6QR = 152.6QR = 152.6LM = 191LM = 191MN = 191MN = 191PQRLMN
Equal intercept theorem: if three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on every transversal. So when PQ = QR, then LM = MN, whatever the slope of the transversals or the spacing of the parallels.

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Selina ICSE: Mid-point Theorem and Its Converse [Including Intercept Theorem]

What this lesson covers

Try to break it

Drag Q and M to slide the parallel lines and tilt the transversals. As long as the three lines stay parallel and cut PQ into equal pieces (PQ = QR), the second transversal is cut into equal pieces too (LM = MN). Try to break the equality on one transversal while keeping it on the other — impossible.

How you build it

Draw three parallel lines cut by two transversals.

  • Draw the first line l (top).
  • Mark a point below l where the second parallel line m will pass.
  • Draw line m parallel to l through the point you just placed.
  • Mark a point below m, about the same distance as m is below l, so the three parallels end up equally spaced.
  • Draw line n parallel to l (and m) through that new point.
  • Line tool: draw a slanted line crossing all three parallels — it meets them at P, Q, R.
  • Line tool: draw another slanted line at a different angle, meeting the parallels at L, M, N — the equally-spaced parallels cut it into equal intercepts too.

The proof, step by step

Prove that equal intercepts on one transversal make equal intercepts on the other.

  • In ΔPQS and ΔQRT, PQ = QR (Given).
  • ∠PQS = ∠QRT and ∠QPS = ∠RQT (Corresponding angles).
  • ΔPQS ≅ ΔQRT by ASA. So PS = QT.
  • PSML and QTNM are parallelograms. So PS = LM and QT = MN.
  • Therefore, LM = MN.

Worked example

Three parallel lines l, m, and n are cut by two transversals. The first transversal makes intercepts of 6 cm and 6 cm. If the second transversal makes an intercept of 4.5 cm on the first gap, what is the length of the intercept on the second gap?

By the Equal Intercept Theorem, equal intercepts on one transversal imply equal intercepts on any other transversal. Since PQ = QR = 6 cm, LM must equal MN. Given LM = 4.5 cm, MN is also 4.5 cm.

  • 4.5 cm — correct
  • 6 cm
  • 3 cm
  • 9 cm
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