Question

using herons formula, calculate the area of the parallelogram to the nearest tenth of a square unit. area = square units

Explanation:

Step1: Recall Heron's formula

Heron's formula for the area of a triangle is $A = \sqrt{s(s - a)(s - b)(s - c)}$, where $s=\frac{a + b + c}{2}$ and $a,b,c$ are the side - lengths of the triangle. A parallelogram can be divided into two congruent triangles. Consider the triangle with side - lengths $a = 5$, $b = 8$, and $c = 11$.
First, calculate the semi - perimeter $s$.
$s=\frac{5 + 8+11}{2}=\frac{24}{2}=12$

Step2: Apply Heron's formula to the triangle

Substitute $s = 12$, $a = 5$, $b = 8$, and $c = 11$ into Heron's formula.
$A_{\triangle}=\sqrt{12(12 - 5)(12 - 8)(12 - 11)}=\sqrt{12\times7\times4\times1}=\sqrt{336}\approx18.33$

Step3: Calculate the area of the parallelogram

Since the parallelogram is composed of two congruent such triangles, $A = 2A_{\triangle}$.
$A = 2\times\sqrt{336}\approx36.7$

Answer:

$36.7$