Question

what does ( c = sqrt{65} ) mean in this problem?
( a = 7 )
( a^{2}+b^{2}=c^{2} )
( 7^{2}+4^{2}=c^{2} )
( 49 + 16 = c^{2} )
( 65 = c^{2} )
( sqrt{65}=c )
( b = 4 )
( c = sqrt{65} )
the length of is ( sqrt{65} ).

Explanation:

Step1: Recall the Pythagorean theorem

In a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the lengths of the legs and \(c\) is the length of the hypotenuse.

Step2: Identify the sides in the given problem

Here, \(a = 7\) and \(b = 4\) are the legs of the right - triangle. After applying the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (i.e., \(7^{2}+4^{2}=c^{2}\), then \(49 + 16=c^{2}\), \(c^{2}=65\), \(c=\sqrt{65}\)), \(c\) represents the hypotenuse.

Answer:

The hypotenuse.