DF bisects <EDG find FG. The diagram is not to scale

Question

EDG x+7 2x+4

Answer 1

Since DF bisects ∠EDG, we can deduce that ∠EDF and ∠FDG are congruent. Therefore, we have:

∠EDF = ∠FDG

Since the sum of the angles in a triangle is 180 degrees, we can set up an equation:

(∠EDF) + (∠FDG) + (∠EDG) = 180

Substituting the given values, we have:

(x+7) + (2x+4) + (∠EDG) = 180

Combining like terms, we get:

3x + 11 + (∠EDG) = 180

Simplifying further:

3x + (∠EDG) = 169

Now, let's find the value of FG based on the angle relations in triangle DFG:

∠EDG and ∠DFG are vertical angles, so they are congruent. Therefore, we have:

∠EDG = ∠DFG

Let FG = y. Then, using the equation for the sum of the angles in triangle DFG, we have:

(∠EDG) + (∠DFG) + (∠DFG) = 180

Substituting the value of ∠EDG from earlier, we get:

∠DFG + ∠DFG + (∠DFG) = 180

Simplifying, we have:

3∠DFG = 180

Dividing both sides by 3, we find:

∠DFG = 60

Since DF bisects ∠EDG, we know that ∠EDF and ∠FDG are congruent. Let each of these angles be θ. Therefore:

∠EDF = ∠FDG = θ

And since ∠DFG = 60, we know that:

2θ + 60 = 180

Subtracting 60 from both sides, we have:

2θ = 120

Dividing both sides by 2, we find:

θ = 60

Now, we can use the value of θ to find the value of x:

∠EDF = x + 7 = 60

Subtracting 7 from both sides, we have:

x = 53

Finally, we can find FG using the value of x:

FG = 2x + 4 = 2(53) + 4 = 106 + 4 = 110

Therefore, FG is equal to 110 units.