Question

using heron’s formula. 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 for triangle area

Heron's formula 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 a triangle. A parallelogram can be divided into two congruent triangles. Let $a = 5$, $b = 8$, $c = 11$.
First, calculate the semi - perimeter $s$:
$s=\frac{5 + 8+11}{2}=\frac{24}{2}=12$

Step2: Calculate the area of one triangle

Substitute $s = 12$, $a = 5$, $b = 8$, $c = 11$ into Heron's formula:
$A_{triangle}=\sqrt{12(12 - 5)(12 - 8)(12 - 11)}=\sqrt{12\times7\times4\times1}=\sqrt{336}=4\sqrt{21}\approx18.33$

Step3: Calculate the area of the parallelogram

Since the area of the parallelogram $A_{parallelogram}=2A_{triangle}$
$A_{parallelogram}=2\times4\sqrt{21}=8\sqrt{21}\approx36.7$

Answer:

$36.7$