What is the length of the unknown side 'a' in the diagram below?

Question

BlackTom AI · Accepted Answer

Using the Pythagorean theorem: in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs.

Relevant knowledge points:
- Identify the right triangle: the small square at the bottom vertex indicates a 90° angle, so the side opposite that angle is the hypotenuse.
- Pythagorean theorem: if the legs are b and c and hypotenuse is h, then b^2 + c^2 = h^2.

Step-by-step reasoning:
1. Label the triangle vertices so the right angle is at C. The side BC (from the right angle to the top vertex) is 80 m, the side AC (from the right angle to the left vertex) is a m, and the side AB (opposite the right angle) is 82 m.
2. Since AB is opposite the right angle, AB = 82 m is the hypotenuse. Apply the Pythagorean theorem:
   a^2 + 80^2 = 82^2.
3. Compute the squares:
   80^2 = 6400,
   82^2 = 6724.
4. Solve for a^2:
   a^2 = 82^2 − 80^2 = 6724 − 6400 = 324.
   Optionally use difference of squares: (82−80)(82+80) = 2×162 = 324.
5. Take the positive square root (lengths are positive):
   a = sqrt(324) = 18 m.

Why the correct answer is correct:
- The calculation directly satisfies Pythagoras for the given triangle: 18^2 + 80^2 = 324 + 6400 = 6724 = 82^2, so a = 18 m.

Why other choices are incorrect:
- 20 m: 20^2 = 400 → 400 + 6400 = 6800 ≠ 6724.
- 17 m: 17^2 = 289 → 289 + 6400 = 6689 ≠ 6724.
- 15 m: 15^2 = 225 → 225 + 6400 = 6625 ≠ 6724.

Conclusion: The length a = 18 m, which matches the provided answer.

What is the length of the unknown side 'a' in the diagram below?单项选择题

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