Verified Answer :

  • The Pythagorean theorem states that in a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: \(a^2 + b^2 = c^2\).
  • To solve for the hypotenuse, rearrange the formula to \(c = \sqrt{a^2 + b^2}\).
  • If numerical values are provided for legs \(a\) and \(b\), calculate their squares, sum them, and take the square root of the sum.
  • The final result for the hypotenuse \(c\) is \(\boxed{\sqrt{a^2 + b^2}}\).

Explanation

  1. Understand the Pythagorean Theorem and the formula for the hypotenuse.

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. For any right-angled triangle with legs of lengths \(a\) and \(b\) and a hypotenuse of length \(c\), the relationship is defined by the equation:
\[a^2 + b^2 = c^2\]
To find the value of the hypotenuse \(c\), we must isolate \(c\) by taking the square root of both sides of the equation. Since the length \(c\) must be positive, we only consider the positive square root:
\[c = \sqrt{a^2 + b^2}\]
This formula allows for the calculation of the hypotenuse given the lengths of the two perpendicular legs. If specific numerical values for \(a\) and \(b\) were provided, one would first square each value, calculate their sum, and then determine the square root of that sum to find \(c\). Finally, as requested, the result should be rounded to the nearest hundredth.

  1. Procedure for calculating the hypotenuse.

To apply this theorem in practice:

  1. Identify the lengths of the two legs \(a\) and \(b\).
  2. Calculate the squares of these lengths: \(a^2\) and \(b^2\).
  3. Sum the squares: \(S = a^2 + b^2\).
  4. Take the square root of the sum: \(c = \sqrt{S}\).
  5. If the resulting value is not an integer, round it to the nearest hundredth by examining the third decimal digit. If it is 5 or greater, round up the second digit; otherwise, round down.

Without specific values for \(a\) and \(b\), the length of the hypotenuse is expressed as:
\[c = \sqrt{a^2 + b^2}\]
This represents the general solution for any right-angled triangle where the legs are known.

Examples

The Pythagorean theorem is incredibly useful in everyday life, such as when you need to calculate the length of a ladder required to reach a specific height on a wall. For example, if you place the base of a ladder \(a = 3\) meters from a wall and the top of the ladder must reach a height of \(b = 4\) meters, you can calculate the necessary ladder length \(c\) using \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) meters. This ensures you buy the correct length of ladder for your task.

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