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Zorluk: OrtaPythagorean Theorem Word Problems

A military training field is rectangular, measuring 160160 yards wide and 300300 yards long. A recruit must travel from one corner of the field to the opposite corner. If the recruit runs directly across the diagonal instead of moving along the outer edges of the two sides, how many yards does the recruit save?

Cevap: 120 yards

Cevap

120 yards
The route along the edges is the sum of the width and length (160+300=460160 + 300 = 460 yards). The diagonal route is the hypotenuse of the right triangle formed by these sides, calculated as 1602+3002=340\sqrt{160^2 + 300^2} = 340 yards. The distance saved is 460−340=120460 - 340 = 120 yards.

Adım Adım Çözüm

1
Calculate the total distance if traveling along the two outer edges of the rectangular field.
160+300=460160 + 300 = 460 yards.
This establishes the longer route distance to compare against the diagonal shortcut.
2
Set up the Pythagorean theorem to find the length of the diagonal route.
1602+3002=c2160^2 + 300^2 = c^2
The rectangular field forms a right triangle where the width and length are the legs, and the diagonal is the hypotenuse.
3
Solve for the diagonal length (cc).
c=115600=340c = \sqrt{115600} = 340 yards.
This gives the exact distance of the direct route.
4
Calculate the difference between the two routes.
460−340=120460 - 340 = 120 yards.
Subtracting the shorter diagonal route from the longer perimeter route yields the exact distance saved.

Anahtar Kavram

Applying the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find a direct diagonal distance, and comparing it to the linear distance along the edges.
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Pythagorean Theorem Word Problems

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