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The lengths of the sides of a triangle are 7​, 5​, and 8. Classify the triangle as​ acute, right, or obtuse.

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Part 1
Choose the correct type of triangle below.

1 answer

To classify the triangle with sides of lengths 7, 5, and 8, we can use the relationships defined by the Pythagorean theorem.

  1. Calculate the squares of the sides:

    • \( a^2 = 7^2 = 49 \)
    • \( b^2 = 5^2 = 25 \)
    • \( c^2 = 8^2 = 64 \)

  2. Determine the relationship among the squares:

    • Arrange the sides such that \( c \) is the longest side. Here, \( c = 8 \), \( a = 7 \), and \( b = 5 \).
    • Evaluate \( a^2 + b^2 \) and compare it to \( c^2 \):
      • \( a^2 + b^2 = 49 + 25 = 74 \)
      • \( c^2 = 64 \)

  3. Compare \( a^2 + b^2 \) to \( c^2 \):

    • Since \( 74 > 64 \), we have \( a^2 + b^2 > c^2 \).

Given the relationship, we can conclude:

  • If \( a^2 + b^2 > c^2 \), the triangle is acute.
  • If \( a^2 + b^2 = c^2 \), the triangle is right.
  • If \( a^2 + b^2 < c^2 \), the triangle is obtuse.

Since \( a^2 + b^2 > c^2 \), the triangle with sides 7, 5, and 8 is classified as acute.