Grade 10 — Mathematics 10-3 › T3U06 Similarity of Polygons

Grade 10 › T3U06 Similarity of Polygons › L2
GRADE 10 · ALBERTAOUTCOME AB-MAT10-3-G3

L2 · Scale, indirect measurement, and nested right triangles

Learning goal

“I can use similar polygons to measure what I cannot reach - heights from shadows, distances from scale drawings - and I can explain why two right triangles that share an acute angle are always similar (the fact trigonometry is built on).”

Indirect measurement — finding a length you cannot measure directly by measuring a similar, reachable situation and using the common ratio (shadows, scale drawings, sighting triangles).
Nested right triangles — two right triangles sharing one acute angle, one drawn inside the other with the same vertex at that angle; the smaller is a scaled copy of the larger.
Proportion — a statement that two ratios are equal, e.g., 3/5 = 12/20; used to solve for one unknown length.
Scale drawing — a drawing in which every length is the actual length times one fixed scale (e.g., 1 cm represents 4 m).
Worked example

A 2 m pole casts a shadow 1.2 m long. At the same moment, a tree casts a shadow 9 m long. How tall is the tree? Step 1 (why similar): the sun's rays hit both at the same angle, and both objects stand at 90 degrees to the ground - the two shadow triangles have matching angles, so they are similar. The RATIO table is legitimate. Step 2 (proportion): height/shadow must match: 2/1.2 = h/9. Step 3 (solve): h = 9 x (2/1.2) = 9 x 1.6667 = 15 m. Step 4 (reasonableness): the tree's shadow (9 m) is much longer than the pole's (1.2 m), so the tree must be much taller than 2 m - 15 m fits. Say it plainly: indirect measurement = build (or spot) a similar triangle you CAN measure, then let the common ratio carry the length across.

Check yourself — explain it

The shadow method works because...

Paired problem — example still visible

A 1.5 m fence post casts a 1 m shadow. A building casts a 24 m shadow at the same time. Building height? -> 36 m (h = 24 x 1.5/1 = 36)

Faded completion — last step is yours

"Scale drawing, 1 cm represents 4 m. A wall drawn 6.5 cm is actually 6.5 x ___ = ___ m." -> 4; 26 m

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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