Geometry

Question Analysis

This question involves applying Pythagorean inequality theorems. The main focus is to determine the type of triangle (acute, right, or obtuse) based on the lengths of its sides by comparing the sum of the squares of the two shorter sides with the square of the longest side.

 

Key Concept Explanation

1. Pythagorean Theorem: For a right triangle with side lengths , , and (where is the longest side), .

2. Pythagorean Inequality for Acute Triangles: If  > , where is the longest side of a triangle with side lengths , , and , then the triangle is acute.

3. Pythagorean Inequality for Obtuse Triangles: If  < , where is the longest side of a triangle with side lengths , , and , then the triangle is obtuse.

 

Step - by - Step Solution

1. Identify the longest side:

Given side lengths are 7, 8, and 10. The longest side , and the two shorter sides are and .

2. Calculate the squares of the sides:

, , and

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