Verified Answer :

  • Identify the sides: \(a=39\), \(b=80\), and \(c=89\).
  • Calculate the sum of the squares of the shorter sides: \(39^2 + 80^2 = 1521 + 6400 = 7921\).
  • Calculate the square of the longest side: \(89^2 = 7921\).
  • Conclude that the sides form a right triangle since \(7921 = 7921\), corresponding to option B.

Explanation

  1. Analyze the condition for a right triangle.

A triangle with sides of lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side) is a right triangle if and only if it satisfies the Pythagorean theorem: \(a^2 + b^2 = c^2\). We are given the side lengths 39, 80, and 89. First, identify the longest side, which is \(c = 89\). The two shorter sides are \(a = 39\) and \(b = 80\). Our goal is to determine if \(39^2 + 80^2 = 89^2\).

  1. Perform calculations and compare the results.

Calculate the squares of the two shorter sides:
\(a^2 = 39^2 = 1521\)
\(b^2 = 80^2 = 6400\)
The sum of these squares is:
\(1521 + 6400 = 7921\)
Next, calculate the square of the longest side:
\(c^2 = 89^2 = 7921\)
Comparing the two, we see that \(7921 = 7921\), which means \(a^2 + b^2 = c^2\).

  1. Draw the final conclusion.

Since the sum of the squares of the shorter sides is exactly equal to the square of the longest side (\(7921 = 7921\)), the given lengths 39, 80, and 89 satisfy the Pythagorean theorem. Therefore, they form the sides of a right triangle. This corresponds to option B.

Examples

The Pythagorean theorem is essential in construction and navigation. For example, a carpenter can ensure a wall corner is perfectly square by measuring 39 inches along one wall, 80 inches along the adjacent wall, and checking if the diagonal distance between the two marked points is exactly 89 inches. If the measurement is correct, the corner is a 90-degree angle.

Related Questions