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Shadows

Part of the Triangles section of Coddy's Geometry journey. Lesson 46 of 60.

Similar triangles measure things you cannot reach. At one moment the sun's rays all come in at the same angle. A post and a tree both stand upright, at right angles to the ground. So the post, its shadow and the ray over its top make a triangle with the same angles as the tree, its shadow and its ray: the triangles are similar by AA.

$A 1 m post with a 2 m shadow beside a tree with a 30 m shadow$

Here a 1 m post casts a 2 m shadow and the tree casts 30 m. The tree's shadow is 15 times the post's, so the tree is 15 times as tall: $\frac{h}{1} = \frac{30}{2}$ , and h = 15 m. Height matches height and shadow matches shadow.

Challenge

A 2 m post PQ casts a 3 m shadow QA. At the same moment a tree UT casts a 15 m shadow TB. Find the height of the tree, UT, then press Check.

Try it yourself

$Find the marked part of the figure$

Find the marked part of the figure

All lessons in Triangles

1The Triangle Inequality

Can Three Sticks Meet?Exactly Reaching The Range of the Third Side Recap: The Third Side

4Area of a Triangle

Half the Rectangle Height Is Not a Side Right Triangle Area Finding the Height Any Side Can Be the Base Recap: Area

7Is It Right?

The Converse Testing a Triangle Acute or Obtuse?Sorting by the Test Recap: Right or Not

10Scale in the World

ShadowsAcross the River Nested Triangles A Side with an Expression Recap: Scale in the World

2Sides and Names

Sorting by Sides The Longest Side Ticks into Equations Recap: Sorting Triangles

5The Pythagorean Theorem

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole Recap: The Hypotenuse

8Special Right Triangles

The 45-45-90 Triangle Legs from the Hypotenuse The 30-60-90 Triangle All Three Sides Which Triangle Is It?Recap: Special Triangles

11Two Tools at Once

Height of an Isosceles Area from the Height The Equilateral Triangle Right Triangle Perimeter Recap: Two Tools