Question

finding the area of a triangle
what information relevant to calculating area do we have available for this triangle?
which method should we use to calculate the area for this triangle?
what is the area of this triangle calculated to the nearest hundredth of a square unit?

Explanation:

Step1: Identify given side lengths

We know all three sides: $DF=7$, $FE=10$, $ED=12$.

Step2: Use Heron's formula (preliminary step)

First calculate semi-perimeter $s$:
$$s=\frac{7+10+12}{2}=\frac{29}{2}=14.5$$

Step3: Apply Heron's formula for area

Area $A=\sqrt{s(s-a)(s-b)(s-c)}$, where $a=7, b=10, c=12$:
$$A=\sqrt{14.5(14.5-7)(14.5-10)(14.5-12)}$$
$$A=\sqrt{14.5\times7.5\times4.5\times2.5}$$
$$A=\sqrt{14.5\times7.5\times11.25}$$
$$A=\sqrt{14.5\times84.375}$$
$$A=\sqrt{1223.4375}$$

Answer:

1. Relevant information: All three side lengths of the triangle: $DF=7$, $FE=10$, $ED=12$
2. Method to use: Heron's Formula (since we know all three sides of the triangle)
3. Calculated area: $34.98$ square units