Question

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 available side - lengths

We have side - lengths \(DE = 12\), \(EF=10\), and \(DF = 7\).

Step2: Choose the Heron's formula

Since we know all three side - lengths of the triangle, we use Heron's formula. First, find the semi - perimeter \(s=\frac{a + b + c}{2}\), where \(a = 12\), \(b = 10\), \(c = 7\). So \(s=\frac{12 + 10+7}{2}=\frac{29}{2}=14.5\).

Step3: Calculate the area using Heron's formula

The area \(A=\sqrt{s(s - a)(s - b)(s - c)}\). Substitute \(s = 14.5\), \(a = 12\), \(b = 10\), \(c = 7\) into the formula: \(A=\sqrt{14.5(14.5 - 12)(14.5 - 10)(14.5 - 7)}=\sqrt{14.5\times2.5\times4.5\times7.5}=\sqrt{14.5\times84.375}=\sqrt{1223.4375}\approx34.98\).

Answer:

The information relevant to calculating area: lengths of all three sides.
The method: Heron's formula.
The area: \(34.98\) square units.