Question

find the area of the triangle.
a 14
b 6
c 19

Explanation:

Step1: Calculate semi-perimeter $s$

First, find the semi-perimeter of the triangle using the formula $s=\frac{a+b+c}{2}$.

$$ s=\frac{14+6+19}{2}=\frac{39}{2}=19.5 $$

Step2: Apply Heron's Formula

Use Heron's formula $Area=\sqrt{s(s-a)(s-b)(s-c)}$ to find the area.

$$ Area=\sqrt{19.5(19.5-14)(19.5-6)(19.5-19)} $$

Step3: Simplify the terms inside the root

Calculate each factor inside the square root.

$$ Area=\sqrt{19.5\times5.5\times13.5\times0.5} $$

Step4: Compute the product inside the root

Multiply the values together.

$$ 19.5\times5.5=107.25, \quad 13.5\times0.5=6.75, \quad 107.25\times6.75=723.9375 $$

Step5: Calculate the square root

Find the square root of the product.

$$ Area=\sqrt{723.9375}\approx26.91 $$

Answer:

$\approx26.91$ (or exactly $\sqrt{723.9375}$)