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Triangles
Introduction
Similar Figures
Practice: Recognising Similar Figures
Similarity of Triangles
Practice: The Basic Proportionality Theorem
Criteria for Similarity of Triangles
Practice: Similarity Criteria in Action
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TrianglesIntroduction

Reading time: ~10 min

It is the morning of 15 August, and Aman's school needs a new rope for the flagpole before the assembly. Nobody knows how tall the pole is, and nobody is allowed to climb it. The PT teacher hands Aman a measuring tape: "You can't reach the top of the pole — but you can reach its shadow."

The pole's shadow measures 10 m. Aman is 1.5 m tall, and in the same sunshine his own shadow measures 2 m.

Find the height of the pole without climbing it. The faint dashed lines are the morning sunlight: every ray comes in at the same slant. Drag the height h of the pole until the ray from its top to the tip of its shadow slants exactly like the sunlight that falls on Aman.

Height of the pole: h = ${h} m

Look at what you have made: two right triangles — Aman with his shadow, and the pole with its shadow — with exactly the same shape but very different sizes. Because the shapes match, every height is the same multiple of its shadow. For Aman, 1.5 \div 2 = 0.75, so the pole is 10 \times 0.75 = m tall.

In Class IX you studied congruent figures: two figures are when they have the same shape and the same size. Aman's triangle and the pole's triangle are not congruent. They are similar figures: the same shape, but not necessarily the same size. This chapter is about such figures, and above all about similar triangles.

Aman's trick is far older than his school. How do you think the height of Mount Everest, or the distance to the Moon, was first found? Nobody stretched a tape to the top of Everest. These lengths were worked out by indirect measurement, and indirect measurement rests entirely on the similarity of figures. You will use it again in Example 7 of this chapter, and in the chapters on trigonometry.

Sina