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Zorluk: OrtaPythagorean Theorem and Special Right Triangles

A surveyor is mapping a triangular park. Starting at point AA, she walks due east for 8080 meters to point BB. She then turns 120∘120^\circ to her left and walks in a straight line to point CC, which is located directly north of point AA. What is the straight-line distance, in meters, from point BB to point CC?

  1. A
    4040
  2. B
    80280\sqrt{2}
  3. C
    80380\sqrt{3}
  4. 160160Cevap
  5. E
    200200

Cevap

The straight-line distance from point BB to point CC is 160160 meters.
The surveyor's movement forms a 30∘-60∘-90∘30^\circ\text{-}60^\circ\text{-}90^\circ right triangle where the side opposite the 30∘30^\circ angle is AB=80AB = 80 meters. The hypotenuse BCBC represents the distance from BB to CC. In a 30∘-60∘-90∘30^\circ\text{-}60^\circ\text{-}90^\circ triangle, the hypotenuse is exactly twice the length of the side opposite the 30∘30^\circ angle. Thus, the distance is 2×80=1602 \times 80 = 160 meters.

Adım Adım Çözüm

1
Determine the orientation and angles of the path.
A right triangle ABCABC with a right angle at vertex AA.
Since the path from AA to BB goes due east, and CC is directly north of AA, the angle ∠A\angle A is exactly 90∘90^\circ.
2
Calculate the interior angle at vertex BB.
∠B=60∘\angle B = 60^\circ and ∠C=30∘\angle C = 30^\circ.
The surveyor turns 120∘120^\circ to the left from the extension of the eastward segment ABAB. The interior angle is the supplement: 180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ. The sum of angles in a triangle is 180∘180^\circ, so the angle at CC is 180∘−(90∘+60∘)=30∘180^\circ - (90^\circ + 60^\circ) = 30^\circ.
3
Use special right triangle ratios to find the hypotenuse.
The distance BC=160BC = 160 meters.
In a 30∘-60∘-90∘30^\circ\text{-}60^\circ\text{-}90^\circ triangle, the sides are in the ratio 1:3:21 : \sqrt{3} : 2. The side opposite the 30∘30^\circ angle is AB=80AB = 80 meters. The hypotenuse BCBC is twice the length of this side: 2×80=1602 \times 80 = 160 meters.

Anahtar Kavram

Ratios of a 30∘-60∘-90∘30^\circ\text{-}60^\circ\text{-}90^\circ special right triangle
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Konular

Pythagorean Theorem and Special Right Triangles

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