Question

given the following side lengths of a triangle, use the pythagorean theorem to determine whether the triangle is a right triangle.
( a = 55 ) in
( b = 77 ) in
( c = 78 ) in

show your work here

hint: to add the square root symbol (( sqrt{} )), type sqrt

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse, the longest side). First, identify the longest side. Here, \(c = 78\) in is the longest. Now calculate \(a^{2}+b^{2}\) and \(c^{2}\).

Step2: Calculate \(a^{2}\)

\(a = 55\), so \(a^{2}=55^{2}=55\times55 = 3025\)

Step3: Calculate \(b^{2}\)

\(b = 77\), so \(b^{2}=77^{2}=77\times77 = 5929\)

Step4: Calculate \(a^{2}+b^{2}\)

\(a^{2}+b^{2}=3025 + 5929=8954\)

Step5: Calculate \(c^{2}\)

\(c = 78\), so \(c^{2}=78^{2}=78\times78 = 6084\)

Step6: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(8954
eq6084\), the triangle does not satisfy the Pythagorean theorem.

Answer:

The triangle with side lengths \(a = 55\) in, \(b = 77\) in, and \(c = 78\) in is not a right triangle.