Answered

The three side lengths of a triangle are given in meters: $12, 13, \sqrt{25}$.



Is a right triangle formed? Explain your reasoning.

Verified Answer :

  • Identify the side lengths as 5 m, 12 m, and 13 m by simplifying \(\sqrt{25}\).
  • Verify that these side lengths satisfy the triangle inequality, confirming a valid triangle exists.
  • Apply the Pythagorean theorem \(a^2 + b^2 = c^2\) with \(a=5, b=12, c=13\).
  • Conclude that the sides form a right triangle since \(5^2 + 12^2 = 13^2\) is true (\(25 + 144 = 169\)).
  • Final answer: \(\boxed{\text{Yes, these three side lengths form a right triangle because } 5^2 + 12^2 = 13^2}\).

Explanation

  1. Analyze side lengths and verify triangle existence.

The problem asks whether a triangle with side lengths 12, 13, and \(\sqrt{25}\) meters forms a right triangle.
First, we simplify the side lengths. The given side lengths are 12 m, 13 m, and \(\sqrt{25}\) m. Since \(\sqrt{25} = 5\), the side lengths are 5 m, 12 m, and 13 m.
Next, we verify the triangle inequality to ensure these sides can form a triangle. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Checking the conditions:

  1. \(5 + 12 > 13 \implies 17 > 13\) (True)
  2. \(5 + 13 > 12 \implies 18 > 12\) (True)
  3. \(12 + 13 > 5 \implies 25 > 5\) (True)

Since all conditions are satisfied, these lengths form a valid triangle.

  1. Apply the Pythagorean theorem to check for a right triangle.

A triangle is a right triangle if and only if the sum of the squares of its two shorter sides equals the square of the longest side (the hypotenuse), according to the Pythagorean theorem: \(a^2 + b^2 = c^2\).
In our case, the sides are \(a = 5\), \(b = 12\), and \(c = 13\).
Calculate the squares of the lengths:

  • \(a^2 = 5^2 = 25\)
  • \(b^2 = 12^2 = 144\)
  • \(c^2 = 13^2 = 169\)

Now, calculate the sum of the squares of the two shorter sides:

  • \(a^2 + b^2 = 25 + 144 = 169\)

Comparing this to the square of the longest side:

  • \(c^2 = 169\)

Since \(a^2 + b^2 = c^2\) (\(169 = 169\)), the side lengths satisfy the Pythagorean theorem. Therefore, the triangle is a right triangle.

Examples

The Pythagorean theorem is essential in construction and architecture. For instance, a carpenter building a rectangular frame needs to ensure the corners are exactly 90 degrees. By measuring a triangle with sides in the ratio 5:12:13 (such as 50 cm, 120 cm, and 130 cm), the carpenter can verify that the corner is a perfect right angle, ensuring the structure is square and stable.

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