Question

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.

Answer 1

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**.