Geometry

Question Analysis

Given the perimeter of a triangle is 30 and side lengths are consecutive integers. First, find the side lengths, then use Heron's formula to calculate the area as all side lengths will be known.

 

Key Concept Explanation

1. Heron's Formula: The area  of a triangle with side lengths , ,  is , where  (semi-perimeter).

2. Consecutive Integers: If the first integer is , the next two consecutive integers are  and . We'll use this property to find the side lengths of the triangle.

 

Step-by-Step Solution

1. Find the side lengths:

Let the side lengths be , , . Since the perimeter , we have .

Combining like terms gives . Subtracting 3 from both sides:

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