Question

goal: we want to find the area of any triangle abc with known sides a, b, and c. fact: we know that the area of a triangle is given by $a=\frac{1}{2}bh$, where b is the base and h is the height. however, in a situation where we cannot easily determine the height, we can find the area of a triangle by using: herons area formula $a = sqrt{s(s - a)(s - b)(s - c)}$ where $s=\frac{a + b + c}{2}$ task #1 - 1: use herons formula to find the exact area of triangle abc with sides: a = 5, b = 12, c = 13 task #1 - 2: use herons formula to find the area of triangle abc with sides: a = 10, b = 12, c = 16 and approximate your answer to 3 significant figures.

Explanation:

Step1: Calculate semi - perimeter for Task #1 - 1

First, find $s$ using $s=\frac{a + b + c}{2}$. For $a = 5$, $b = 12$, $c = 13$, we have $s=\frac{5+12 + 13}{2}=\frac{30}{2}=15$.

Step2: Calculate area for Task #1 - 1

Then, use Heron's formula $A=\sqrt{s(s - a)(s - b)(s - c)}$. Substitute $s = 15$, $a = 5$, $b = 12$, $c = 13$: $A=\sqrt{15(15 - 5)(15 - 12)(15 - 13)}=\sqrt{15\times10\times3\times2}=\sqrt{900}=30$.

Step3: Calculate semi - perimeter for Task #1 - 2

For $a = 10$, $b = 12$, $c = 16$, find $s=\frac{10+12 + 16}{2}=\frac{38}{2}=19$.

Step4: Calculate area for Task #1 - 2

Use Heron's formula: $A=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{19(19 - 10)(19 - 12)(19 - 16)}=\sqrt{19\times9\times7\times3}=\sqrt{3591}\approx59.9$.

Answer:

Task #1 - 1: 30
Task #1 - 2: 59.9