Chapter 6 — Triangles
A photograph and its enlargement, a map and the land it shows, a pole and the tower beside it: figures with the same shape but different sizes. Learn what "same shape" means exactly, and the tests that prove two triangles are similar.
On a sunny morning a $1$ m stick, held upright, casts a shadow $1.6$ m long. At the same moment a tree beside it casts a shadow $12.8$ m long. How tall is the tree? You cannot climb it with a tape measure, yet the answer is easy: the shadow of the tree is $12.8\div1.6=8$ times as long as the stick's shadow, so the tree is $8$ times as tall as the stick — $8$ m. The reason this works is that the stick, its shadow and the sun's ray make a small triangle with exactly the same shape as the large triangle made by the tree, its shadow and the ray. One is just an enlargement of the other.
That idea — same shape, different size — is called similarity, and it is behind maps and scale drawings, photographs and projector screens, model aeroplanes, and the way surveyors measure towers and rivers they cannot reach. In this chapter you first make "same shape" precise for polygons. Then you study triangles, where a surprisingly short list of facts is enough: a line drawn parallel to one side of a triangle cuts the other two sides in the same ratio (the Basic Proportionality Theorem), and two triangles are similar as soon as two angles, or three ratios of sides, or one angle and the two sides around it, match up.
- say when two polygons are similar, and explain why equal angles alone (or proportional sides alone) are not enough for polygons with more than three sides;
- write a similarity statement such as $\triangle ABC\sim\triangle PQR$ with the vertices in the correct order, and read off the equal angles and the ratios of corresponding sides;
- state and prove the Basic Proportionality Theorem, and use it, and its converse, to find lengths and to decide whether two lines are parallel;
- test two triangles for similarity with the AA, SSS and SAS criteria and choose the right criterion for the data given;
- solve problems on shadows, poles, ladders and trapeziums by setting up proportions from similar triangles.
Sections in this chapter
Similar Figures
What it means for two shapes to be similar, why both the angles and the sides matter, and how to name corresponding vertices in a similarity statement.
The Basic Proportionality Theorem
A line parallel to one side of a triangle divides the other two sides in the same ratio — with its proof, its converse and its uses.
Criteria for Similarity of Triangles
The AA, SSS and SAS tests that prove two triangles are similar, and how to use them in applications.