L1 · Ratios born from similar right triangles: patterns
“I can show that for one FIXED acute angle, the ratios opposite/adjacent, opposite/hypotenuse and adjacent/hypotenuse are the same in every right triangle that contains that angle - and I can use those constant ratios to predict sides of bigger or smaller triangles in the family.”
Three right triangles share the acute angle at A, drawn nested: small (opposite 3 cm, adjacent 4 cm, hypotenuse 5 cm), medium (6, 8, 10), large (9, 12, 15). Step 1 (compute the ratio in the small): opposite/adjacent = 3/4 = 0.75. Step 2 (medium): 6/8 = 0.75. Step 3 (large): 9/12 = 0.75. Say it plainly: the triangles grew by scale factors 2 and 3, and BOTH sides in the ratio grew by the same factor - so the ratio cannot change. This quotient, opposite/adjacent for angle A, is a property of the ANGLE, not of the triangle's size. In L2 it gets its name: the tangent of A. Ratio table for angle A: opposite/adjacent - 3/4 = 0.75, 6/8 = 0.75, 9/12 = 0.75.
The ratio stayed 0.75 in all three triangles because...
A fourth triangle in the family has adjacent side 20 cm. Its opposite side? -> 15 cm (0.75 x 20; also 3-4-5 scaled by 5)
"In the family, opposite/adjacent = ___. A triangle with opposite side 24 cm has adjacent side 24 / ___ = ___ cm." -> 0.75; 0.75; 32 cm
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.