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Posttest The Real Number System

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Two sides of a right triangle measure 5 Inches and 6 Inches, and the third side measures √61 Inches.

The length of the third side is closest to Choose an answer Inches. The perimeter of the triangle is closest to

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Verified Answer :

  • Identify that the triangle satisfies the Pythagorean theorem with side lengths 5, 6, and \(\sqrt{61}\).
  • Approximate \(\sqrt{61} \approx 7.8\) by observing it lies between \(7^2=49\) and \(8^2=64\).
  • Calculate the perimeter by summing the sides: \(5 + 6 + 7.8 = 18.8\).
  • The final results are a third side length of \(\boxed{7.8}\) and a perimeter of \(\boxed{18.8}\).

Explanation

  1. Problem Analysis

The triangle is given as a right triangle with side lengths 5 inches, 6 inches, and \(\sqrt{61}\) inches. We first verify that these sides satisfy the Pythagorean theorem: \(a^2 + b^2 = c^2\). Here, \(5^2 + 6^2 = 25 + 36 = 61\), and \((\sqrt{61})^2 = 61\). Since \(61 = 61\), the triangle is indeed a right triangle. Our goal is to determine the numerical approximation for \(\sqrt{61}\) and the total perimeter of the triangle to the nearest tenth.

  1. Calculating the Third Side Length

To find the length of the third side, \(\sqrt{61}\), we look at the perfect squares around 61, which are \(7^2 = 49\) and \(8^2 = 64\). Since 61 is much closer to 64 than 49, the value of \(\sqrt{61}\) will be slightly less than 8. Calculating this precisely gives \(\sqrt{61} \approx 7.8102\). Rounding to the nearest tenth, we get 7.8.

  1. Calculating the Perimeter

The perimeter \(P\) of a triangle is the sum of its three side lengths. Thus, the perimeter is:
\[P = 5 + 6 + \sqrt{61} = 11 + \sqrt{61}\]
Using our approximation \(\sqrt{61} \approx 7.8102\), the perimeter is approximately:
\[P \approx 11 + 7.8102 = 18.8102\]
Rounding this to the nearest tenth, we get 18.8.

  1. Final Result

The length of the third side is approximately 7.8 inches and the perimeter is approximately 18.8 inches.
The final answers are: third side length is \(\boxed{7.8}\) and the perimeter is \(\boxed{18.8}\).

Examples

Understanding how to approximate square roots is essential in various fields, such as construction and interior design. For example, if you are measuring the length of a diagonal brace for a shelf that is 5 feet by 6 feet, you can quickly estimate the length of the brace using the Pythagorean theorem, \(\sqrt{5^2 + 6^2} = \sqrt{61} \approx 7.8\) feet. This approximation allows for efficient planning of materials without needing a calculator for every small adjustment.

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