Suppose that
𝑎
2
+
𝑏
2
=
𝑐
2
and
𝑎
=
5
and
𝑏
=
12
. Which of the following shows the correct answer for
𝑐
?

Question

Suppose that 
𝑎
2
+
𝑏
2
=
𝑐
2
 and 
𝑎
=
5
 and 
𝑏
=
12
.  Which of the following shows the correct answer for 
𝑐
?

BlackTom AI · Accepted Answer

By the Pythagorean theorem, c^2 = a^2 + b^2.

1. Relevant knowledge points and concepts
- The Pythagorean theorem relates the sides of a right triangle: a^2 + b^2 = c^2, where c is the hypotenuse (or more generally the unknown whose square equals the sum of the squares).
- Solving an equation of the form c^2 = K requires taking square roots; algebraically this yields two solutions, c = +√K and c = −√K, because both square to K.
- If c represents a geometric length, only the positive root is physically meaningful; algebraically both signs are valid solutions.

2. Complete reasoning process step by step
- Substitute the given values a = 5 and b = 12 into the equation a^2 + b^2 = c^2.
- Compute a^2 and b^2, add them, then solve for c by taking square roots.

3. Calculations with intermediate steps
- a^2 = 5^2 = 25
- b^2 = 12^2 = 144
- a^2 + b^2 = 25 + 144 = 169
- Therefore c^2 = 169
- Taking square roots: c = ±√169 = ±13

4. Why the correct answer is correct
- The algebraic solution to c^2 = 169 is c = ±13 because both 13^2 and (−13)^2 equal 169. This follows directly from the definition of square roots and the Pythagorean relation used with the provided a and b.

5. Why other possible choices would be incorrect
- Choosing only c = 13 omits the algebraic negative root (c = −13), so it is not the complete set of solutions.
- Choosing only c = −13 is incorrect because it omits the positive root and, if c denotes a length, is nonphysical.
- Any other number (e.g., 12, 5, 0, or √169 misinterpreted) does not satisfy c^2 = 169.

6. Conclusion confirming the final answer
- The correct algebraic solutions are c = +13 or −13, matching the provided answer: (d) c = +13 or -13.

Suppose that 𝑎 2 + 𝑏 2 = 𝑐 2 and 𝑎 = 5 and 𝑏 = 12 . Which of the following shows the correct answer for 𝑐 ?单项选择题

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