Question

Triangle RST has sides measuring 22 inches and 13 inches and a perimeter of 50 inches. What is the area of triangle RST? Round to the nearest square inch. (using heron's formula)

Answer 1

$\bf \textit{Heron's Area Formula}\\\\ A=\sqrt{s(s-a)(s-b)(s-c)}\qquad \begin{cases} a=22\\ b=13\\ c=\stackrel{perimeter}{50}-22-13\\ \qquad 15\\ s=\frac{a+b+c}{2}\\ \qquad 25 \end{cases} \\\\\\ A=\sqrt{25(25-22)(25-13)(25-15)}\implies A=\sqrt{25(3)(12)(10)} \\\\\\ A=\sqrt{25(360)}\implies A=\sqrt{9000}\implies A\approx 94.868329805$

Answer 2

The area of the triangle using Heron's formula is 95 square inches.

What is a triangle?

A triangle is a flat geometric figure that has three sides and three angles. The sum of the interior angles of a triangle is equal to 180°. The exterior angles sum up to 360°.

For the given situation,

The triangle sides are a =22 inches and b = 13 inches.

The perimeter of the triangle, P = 50 inches.

Let the third side of the triangle be x.

The formula of perimeter of triangle is

$P=a+b+c$

⇒ $50=22+13+x$

⇒ $x=50-35$

⇒ $x=15$

The Heron's formula of area of triangle is

$A=\sqrt{s(s-a)(s-b)(s-c)}$

where $s=\frac{a+b+c}{2}$

⇒ $s=\frac{22+13+15}{2}$

⇒ $s=25$

Now substitute the above values,

⇒ $A=\sqrt{25(25-22)(25-13)(25-15)}$

⇒ $A=\sqrt{25(3)(12)(10)}$

⇒ $A=\sqrt{9000}$

⇒ $A=94.86$ ≈ $95$

Hence we can conclude that the area of the triangle using Heron's formula is 95 square inches.

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