Final answer:
To find the largest angle in triangle DEF, we can use the Law of Cosines to solve for the angle ∠DEF. Plugging in the given side lengths, we can calculate the cosine of ∠DEF and then take its inverse to find the value of the angle. The largest angle in triangle DEF is approximately 119.2°.
Step-by-step explanation:
To find the largest angle in triangle DEF, we can use the Law of Cosines. According to the law, the square of one side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides and the cosine of the angle between them. In this case, we have:
DF^2 = DE^2 + EF^2 - 2 * DE * EF * cos(∠DEF)
Plugging in the given values, we get:
22^2 = 12^2 + 14^2 - 2 * 12 * 14 * cos(∠DEF)
Simplifying the equation, we have:
484 = 144 + 196 - 336 * cos(∠DEF)
Now we can solve for cos(∠DEF):
cos(∠DEF) = (144 + 196 - 484) / (336 * -1)
cos(∠DEF) ≈ -0.0595
Since cos(∠DEF) is negative, the angle ∠DEF is obtuse. To find the largest angle, we need to take the inverse cosine of cos(∠DEF):
∠DEF ≈ 119.2°
Therefore, the largest angle in triangle DEF is approximately 119.2°.