Question

Find the area of triangle DFG.

A. 5.3 square units
B. 34.3 square units
C. 420.0 square units
D. 424.2 square units

Answer 1

34.3 square unit.

Explanation:

`calculate \: df \: using \: pythagoras \\ theorem \\ df {}^(2) = 8 {}^(2) + 6 {}^(2) \\ df {}^(2) = 64 + 36 \\ df {}^(2) = 100 \\ df = √(100) \\ df = 10 \\ calculating \: for \: the \: area \: of \: \\ angle \: \: dfg \\ using \: heros \: formular \\ \\ area = √(s(s - a)(s - b)(s - c)) \\ s = (a + b + c)/(2) \\ s = (10 + 11 + 7)/(2) = (28)/(2) = 14 \\ area = √(14(14 - 10)(14 - 11)14(14 - 7)) \\ area = √(14(4)(3)(7)) = √(14 * 84) \\ area = √(1176) = 34.2928564 \\ to \: the \: nearest \: tenth \: = 34.3`

Answer 2

• Option B. 34.3 square units is correct!

Explanation :

Here it is stated that, side DE = 8 units, side EF = 6 units, side FG = 7 units and side GD = 11 units. We have to find area of ∆DFG, here we will use heron's formula which is given by:

Area of ∆ = √[s(s – a) (s – b) (s – c)]

Here a, b, and c are sides of ∆. We have;

• b = FG = 7 units
• c = GD = 11 units
• a = DF = ?
• s = semi - perimeter = ?

So firstly lets calculate a i.e DF by using Pythagoras theorem on ∆DEF:

➸ DF² = 8² + 6²

➸ DF² = (8 × 8) + (6 × 6)

➸ DF² = 64 + 36

➸ DF² = 100

➸ DF = √(100)

➸ DF = √(10 × 10)

➸ DF = 10 units

Now, lets calculate s i.e semi - perimeter:

• s = (a + b + c)/2
• s = (10 + 7 + 11)/2
• s = 28/2
• s = 14 units

Now, using heron's formula on ∆DFG to calculate its area:

➸ Area(∆DFG) = √[14(14 – 10) (14 – 7) (14 – 11)]

➸ Area(∆DFG) = √[14(4) (7) (3)

➸ Area(∆DFG) = √(14 × 4 × 7 × 3)

We can write it as;

➸ Area(∆DFG) = √(2 × 2 × 2 × 7 × 7 × 3)

➸ Area(∆DFG) = 2 × 7√(2 × 3)

➸ Area(∆DFG) = 14√(6)

➸ Area(∆DFG) = 14 × 2.449

➸ Area(∆DFG) = 34.28

➸ Area(∆DFG) = 34.3 square units (approx)

• Hence, area of ∆DFG is option B. 34.3 square units.