Question

1. find the perimeter and area of this triangle. dimensions are in meters.

Explanation:

Step1: Calculate the perimeter

Add all side lengths:
$7 + 11 + 13 = 31$

Step2: Calculate semi-perimeter ($s$)

Use $s=\frac{a+b+c}{2}$:
$s = \frac{7+11+13}{2} = \frac{31}{2} = 15.5$

Step3: Apply Heron's formula for area

Use $A=\sqrt{s(s-a)(s-b)(s-c)}$:
$A = \sqrt{15.5(15.5-7)(15.5-11)(15.5-13)}$

Step4: Simplify the expression

Compute each term inside the square root:
$15.5-7=8.5$, $15.5-11=4.5$, $15.5-13=2.5$
$A = \sqrt{15.5 \times 8.5 \times 4.5 \times 2.5}$
$A = \sqrt{\frac{31}{2} \times \frac{17}{2} \times \frac{9}{2} \times \frac{5}{2}} = \sqrt{\frac{31 \times 17 \times 9 \times 5}{16}}$
$A = \frac{3}{4}\sqrt{31 \times 17 \times 5} = \frac{3}{4}\sqrt{2635} \approx 37.4$

Answer:

Perimeter: 31 meters
Area: $\frac{3}{4}\sqrt{2635}$ square meters (or approximately 37.4 square meters)