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Difficulty: MediumRight Triangles and the Pythagorean Theorem

A surveyor stands at point AA on one bank of a straight river, directly across from a tree TT on the opposite bank. She walks along the riverbank to point BB such that the line segment ABAB is perpendicular to ATAT. The distance from AA to BB is 77 meters. She then continues walking in the same direction along the riverbank for another 2525 meters to point CC. If the straight-line distance from point CC to the tree TT is 4040 meters, what is the straight-line distance, in meters, from point BB to the tree TT?

  1. A
    8.75
  2. B
    11.2
  3. 25Answer
  4. D
    31

Answer

25
The correct answer is 25. Since the segments ABAB and BCBC lie on a straight line and are perpendicular to ATAT, triangle TACTAC is a right triangle with leg AC=7+25=32AC = 7 + 25 = 32 meters and hypotenuse TC=40TC = 40 meters. By the Pythagorean theorem, the common leg TATA is equal to 402−322=576=24\sqrt{40^2 - 32^2} = \sqrt{576} = 24 meters. Using the Pythagorean theorem again on the smaller right triangle TABTAB with legs TA=24TA = 24 meters and AB=7AB = 7 meters, the hypotenuse TBTB is equal to 242+72=625=25\sqrt{24^2 + 7^2} = \sqrt{625} = 25 meters.

Step-by-Step Solution

1
Determine the total distance from point AA to point CC along the riverbank.
AC=AB+BC=7+25=32AC = AB + BC = 7 + 25 = 32 meters.
Points AA, BB, and CC lie on a straight line in that order, so their distances add linearly.
2
Apply the Pythagorean theorem to the right triangle TACTAC to find the distance from AA to the tree TT.
TA=TC2−AC2=402−322=1600−1024=576=24TA = \sqrt{TC^2 - AC^2} = \sqrt{40^2 - 32^2} = \sqrt{1600 - 1024} = \sqrt{576} = 24 meters.
Since ATAT is perpendicular to ACAC, triangle TACTAC is a right triangle with hypotenuse TCTC and legs TATA and ACAC.
3
Apply the Pythagorean theorem to the right triangle TABTAB to find the distance from BB to the tree TT.
TB=TA2+AB2=242+72=576+49=625=25TB = \sqrt{TA^2 + AB^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 meters.
Triangle TABTAB is a right triangle with legs TATA and ABAB and hypotenuse TBTB.

Key Concept

Using the Pythagorean theorem in multi-step geometric problems with right triangles sharing a common side.
Estimated Time:1m 30s
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Topics

Right Triangles and the Pythagorean Theorem

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