Answered

Does the following set form the lengths of the sides of a right triangle?

39, 80, 89



Choose the correct answer below.

ANo, because the sum of the squares of the smaller numbers is not equal to the square of the larger number.
BNo, because the square of the sum of any two numbers is not equal to the square of the remaining number.
CYes, because the sum of the squares of the smaller numbers is equal to the square of the larger number.
DYes, because the square of the sum of any two numbers is equal to the square of the remaining number.

Verified Answer :

  • Identify the lengths as \(a=39\), \(b=80\), and the hypotenuse candidate \(c=89\).
  • Apply the Pythagorean theorem by calculating \(a^2 + b^2 = 39^2 + 80^2 = 7921\) and \(c^2 = 89^2 = 7921\).
  • Compare the two results: since they are equal, the set forms a right triangle.
  • Conclude that option C is the correct choice: \(\boxed{C}\).

Explanation

  1. Analysis of the Pythagorean Theorem

To determine if a set of three lengths forms a right triangle, we apply the Pythagorean theorem. This theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), where \(c\) is the length of the longest side, the triangle is a right triangle if and only if:
\[a^2 + b^2 = c^2\]
In our case, the given lengths are \(39\), \(80\), and \(89\). The longest side is \(c = 89\). The two shorter sides are \(a = 39\) and \(b = 80\).

  1. Calculation and Comparison

We first calculate the square of the longest side:
\[c^2 = 89^2 = 7921\]
Next, we calculate the sum of the squares of the two shorter sides:
\[a^2 + b^2 = 39^2 + 80^2\]
\[a^2 + b^2 = 1521 + 6400 = 7921\]
Comparing the two results, we see that \(a^2 + b^2 = 7921\) and \(c^2 = 7921\). Since these values are equal, the condition \(a^2 + b^2 = c^2\) is satisfied.

  1. Conclusion

Because the sum of the squares of the two smaller sides is equal to the square of the longest side, the set of lengths \(39, 80, 89\) forms a right triangle. This corresponds to option C: Yes, because the sum of the squares of the smaller numbers is equal to the square of the larger number.
The final answer is \(\boxed{C}\).

Examples

Carpenters and construction workers frequently use the Pythagorean theorem to ensure that corners of buildings are perfectly square. For instance, if you are building a small rectangular platform, you can measure 39 inches along one edge and 80 inches along the adjacent edge. If you measure the diagonal distance between these two points and find it to be exactly 89 inches, you can be certain that the angle between the two sides is exactly \(90^{\circ}\). This method is often called the "3-4-5 rule" for smaller projects, scaled up to larger dimensions like the 39-80-89 triangle here.

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