Solve for c.
$c = [?]$
Pythagorean Theorem: $a^2 + b^2 = c^2$
Round your answer to the nearest hundredth.
The problem asks to solve for the hypotenuse \(c\) of a right-angled triangle given the lengths of the two other sides, \(a\) and \(b\). The Pythagorean Theorem states that for any right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the relationship between the lengths is given by the equation:
\[a^2 + b^2 = c^2\]
To solve for the hypotenuse \(c\), we take the square root of both sides:
\[c = \sqrt{a^2 + b^2}\]
Since the specific numerical values for \(a\) and \(b\) are not provided, the solution is represented by the formula \(\sqrt{a^2 + b^2}\). If specific values for \(a\) and \(b\) were provided, you would substitute them into this formula, calculate the sum of their squares, and then find the square root. Finally, you would round the resulting value to the nearest hundredth as requested.
To find the hypotenuse \(c\), first identify the lengths of the two legs, \(a\) and \(b\). Then, square each of these lengths independently: \(a^2\) and \(b^2\). Sum these two squares to get \(a^2 + b^2\). Finally, find the square root of this sum: \(\sqrt{a^2 + b^2}\). If the resulting value is not an integer, identify the digit in the thousandths place to determine whether to round up or keep the current value in the hundredths place. The final answer for \(c\) is the value rounded to the nearest hundredth.
The Pythagorean Theorem is incredibly useful in everyday life, such as when you need to calculate the length of a ladder needed to reach a certain height on a wall. If you know how far the base of the ladder is from the wall (\(a\)) and the height the ladder needs to reach (\(b\)), you can calculate the necessary length of the ladder (\(c\)) using \(c = \sqrt{a^2 + b^2}\). For example, if the base is 3 meters from the wall and the height is 4 meters, the ladder length must be \(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) meters.