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 \(a\), \(b\), \(c\), the triangle inequality states \(a + b>c\), \(a + c>b\), and \(b + c>a\).
\(b + c=9 + 5=14>12=a\)
\(a + c=12 + 5=17>9=b\)
\(a + b=12+9 = 21>5=c\)

Step2: Calculate semi - perimeter \(s\)

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: Use Heron's formula

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\).
Substitute \(s = 13\), \(a = 12\), \(b = 9\), \(c = 5\) into the formula:
\(A=\sqrt{13(13 - 12)(13 - 9)(13 - 5)}=\sqrt{13\times1\times4\times8}=\sqrt{416}=4\sqrt{26}\approx20.4\)

Answer:

The given side lengths \(a = 12\), \(b = 9\), \(c = 5\) form a triangle. The area of the triangle is \(4\sqrt{26}\approx20.4\)