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A woman standing 6 feet west of the southwest corner of her house is watching a bird 11 feet up in a tree that is 8 feet south of the southwest corner of the house. When observing the bird using binoculars, she holds the binoculars 5 feet off the ground. What is the approximate distance from the bird to the binoculars?

1 Answer

5 votes

Answer:

11.7 feet

Explanation:

You want the approximate distance from binoculars to a bird if the binoculars are 5 feet off the ground at a point 6 feet west of the corner of the house, and the bird is 11 feet off the ground at a point 8 feet south of the corner of the house.

Triangles

The distance from the woman to the tree is the hypotenuse of a right triangle with legs 6 ft and 8 ft. You will recognize this as a 3-4-5 right triangle with a scale factor of 2. The hypotenuse is 2·5 ft = 10 ft.

The distance from the level of the binoculars to the bird is 11 ft less 5 ft, or 6 feet. Thus the distance from binoculars to bird is the hypotenuse of a right triangle with legs 6 feet and 10 feet. That distance is ...

d = √(6² +10²) = √136 ≈ 11.7 . . . . feet

The approximate distance from the bird to the binoculars is 11.7 feet.

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Additional comment

The same distance formula can be used for the triangle with legs 6 ft and 8 ft:

d = √(6² +8²) = √100 = 10 . . . . feet, as described above.

It is useful to be able to recognize some common Pythagorean triples:

{3, 4,5}, {5, 12, 13}, {7, 24, 25}, {8, 15, 17}

These, and their multiples, are a few of the integer side lengths that form right triangles.

The attached diagram shows an elevation view below and left of the line WT joining the woman's position with the tree in the plan view.

A woman standing 6 feet west of the southwest corner of her house is watching a bird-example-1
User Avihu Turzion
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