Our friend determines whether a triangle with side lengths 3, sqrt(58), and 7 is a right triangle. Let's not look at our friend's solution just yet, and decide if the triangle is a right triangle on our own first. Let's start by recalling the Converse of the Pythagorean Theorem.
Converse of the Pythagorean Theorem
If the equation a^2 + b^2 = c^2 is true for the side lengths of a triangle, then a triangle is a right triangle.
If the equation a^2 + b^2 = c^2 is satisfied, then a and b are the lengths of the legs and c is the length of the hypotenuse. Since the hypotenuse is the longest side in the right triangle, we should always substitute the length of the longest side for c. Let's find the longest side.
3<7
We know that 7 is greater than 3. To compare 7 and sqrt(58), let's rewrite 7 as a square root.
We can rewrite 7 as sqrt(49). Since 49 is less than 58, we know that sqrt(49) < sqrt(58). Therefore, 7 < sqrt(58).
rcl
49 & < & 58
& ⇓ &
sqrt(49) & < & sqrt(58)
& ⇓ &
7 & < & sqrt(58)
The longest side of our triangle is the side with a length of sqrt(58). Now, let's substitute 3 for a, 7 for b, and sqrt(58) for c into the equation a^2 + b^2 = c^2. Then, we will simplify it and see if the equation produces a true statement.
We obtained a true statement. This means that the triangle is a right triangle. Now, let's consider our friend's solution.
Our friend substituted 7 for c, but the side of the length 7 is not the longest side. They should substitute sqrt(58) for c. This means that our friend is not correct.
