Question

1. first determine if the given measurements will make a triangle. if they do, find the area of the triangle. $a = 12,b = 9,c = 5$

Explanation:

Step1: Check triangle inequality

For three side lengths \(a\), \(b\), \(c\) (\(a\geq b\geq c\)), the triangle inequality states \(b + c>a\).
Here \(a = 12\), \(b = 9\), \(c = 5\). Calculate \(b + c=9 + 5=14\). Since \(14>12\), a triangle can be formed.

Step2: Calculate semi - perimeter \(s\)

The formula for the semi - perimeter \(s=\frac{a + b + c}{2}\).
Substitute \(a = 12\), \(b = 9\), \(c = 5\) into the formula: \(s=\frac{12 + 9+5}{2}=\frac{26}{2}=13\).

Step3: Calculate area using Heron's formula

Heron's formula is \(A=\sqrt{s(s - a)(s - b)(s - c)}\).
Substitute \(s = 13\), \(a = 12\), \(b = 9\), \(c = 5\) into the formula:

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Answer:

The given side lengths form a triangle, and the area of the triangle is \(4\sqrt{26}\approx20.4\)