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Zorluk: OrtaTriangles and Pythagorean Theorem

A naval patrol vessel leaves port and sails due South for 10 nautical miles10\text{ nautical miles}. It then changes course and sails due West. After completing this leg of the journey, the vessel's radar indicates it is exactly 26 nautical miles26\text{ nautical miles} in a straight line from the port. How many nautical miles did the vessel travel due West?

  1. A
    1616
  2. 2424Cevap
  3. C
    3636
  4. D
    576576

Cevap

The vessel traveled 2424 nautical miles due West.
The path of the vessel forms a right triangle. The straight-line distance from the port is the hypotenuse (26 nautical miles26\text{ nautical miles}), and the southward distance is one leg (10 nautical miles10\text{ nautical miles}). Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we substitute the known values: 102+b2=26210^2 + b^2 = 26^2. This simplifies to 100+b2=676100 + b^2 = 676. Subtracting 100100 from both sides gives b2=576b^2 = 576. Taking the square root of 576576 yields 24 nautical miles24\text{ nautical miles}.

Adım Adım Çözüm

1
Identify that the vessel's path forms a right triangle.
The southward path (1010) and westward path (xx) are the legs, and the straight-line distance (2626) is the hypotenuse.
South and West directions are perpendicular, creating a 90-degree angle.
2
Set up the Pythagorean theorem equation (a2+b2=c2a^2 + b^2 = c^2).
102+x2=26210^2 + x^2 = 26^2
This formula relates the lengths of the legs to the hypotenuse in a right triangle.
3
Square the known values.
100+x2=676100 + x^2 = 676
To begin isolating the unknown variable xx.
4
Subtract 100100 from both sides of the equation.
x2=576x^2 = 576
To completely isolate the x2x^2 term.
5
Take the square root of both sides.
x=24x = 24
To find the final distance traveled due West.

Anahtar Kavram

Applying the Pythagorean theorem to find the missing leg of a right triangle.
Tahmini Süre:1m 0s
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Konular

Triangles and Pythagorean Theorem

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