ACT-Style: Multi-Step Triangle Problems

ACT Math · Axiom Academy

EXAMPLE ACT-Style: Multi-Step Triangle Problems Chaining a Pythagorean triple, similar triangles, and the area-ratio shortcut in one ACT problem. Triangle ABC has sides AB = 15 , BC = 20 , and AC = 25 . A line parallel to meets at D and at E , with AD = 6 . What is the area of trapezoid BCED ? Triangle ABC (right-angled at B ) with ; the trapezoid BCED is outlined in orange. Nice work — you just chained three separate triangle tools into one ACT answer, exactly the kind of multi-step reasoning the test rewards. Spot the triple: is a scaled , so ABC is right-angled at B and its area is just — no height to hunt for. Parallel line ⟹ similar triangles: because , with scale factor . The area-ratio shortcut: similar triangles with scale factor k have areas in ratio k^2 , so . Result: the trapezoid is the leftover: 150 - 24 = 126 . The k^2 area rule is one of the most powerful shortcuts on ACT geometry — forgetting to square k (which gives 90 ) is exactly the trap answer the test plants for you.

This is the written version of the interactive lesson above. See the full ACT Math course.