Chapter 5.5, Problem 25E

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Chapter
Section

### Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

# A jogger runs along two sides of an open rectangular lot. If the first side of the lot is 200 ft long and the diagonal distance across the lot is 400 ft, what is the measure of the angle formed by the 200 -ft and 400 -ft dimensions? To the nearest foot, how much farther does the jogger run by travelling the two sides of the block rather than the diagonal distance across the lot?

To determine

To find:

The measure of the angle formed by the 200 ft. and 400 ft. dimensions and the distance by which the jogger ran farther by travelling the two sides of the block rather than the diagonal across the lot.

Explanation

Approach:

For a right triangle, for which the measure of the interior angles 30Â°, 60Â°, and 90Â°; if â€˜aâ€™ is the length of measure of the shorter leg; opposite to the angle 30Â°, then the length of the other two sides is given by

Length of the longer leg (opposite to 60Â°) =a3

Length of the hypotenuse (opposite to 90Â°)=2a.

In general

Length of the longer leg =3Ã— (Length of the shorter leg)

Length of the hypotenuse =2Ã— (Length of the shorter leg)

Calculation:

Given,

Jogger runs along two sides of an open rectangular lot, the first side of the lot is 200 ft. long and the diagonal across the lot is 400 ft.

The width of the rectangular lot =200 ft.

The diagonal of the rectangular lot =400 ft.

Let the rectangular lot be named as ABCD with AC as diagonal.

Now, the triangle ABC formed by the diagonal AC of the rectangle is a right triangle with

Right angle at B,

AC=400 ft = Hypotenuse

BC=200 ft.

Here, we find that in âˆ†ABC

AC=2BC

Hypotenuse =2(BC)

But, by the 30-60-90 theorem of a triangle,

30Â°-60Â°-90Â° theorem.

In a right triangle whose angle measure 30Â°, 60Â°, and 90Â°, the hypotenuse has a length equal to twice the length of the shorter leg, and the longer leg is the product of 3 and the length of the shorter leg.

Hypotenuse =2Ã— (Length of the shorter side)

Thus, BC is the shorter side of the right triangle which always lies opposite to 30Â°.

Hence,

mâˆ A=30Â°

Further, AB should be the longer side of the right âˆ†, which should lie opposite to the angle 60Â°

In right âˆ†ABC apply phythagorean theorem to find AB the second side of the rectangular lot

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