To find FG, we need to find the value of n.
Given that angle E and angle G are both 90 degrees, we know that triangle EDG is a right triangle.
Since EF is a side of triangle EDG, it is opposite to angle E, making it the hypotenuse.
Similarly, FG is a side of triangle EDG, it is opposite to angle G, making it one of the legs.
We have EF = n + 9 and FG = 4n - 6.
Since EF is the hypotenuse, we can use the Pythagorean theorem to find its length.
Using the Pythagorean theorem, we have:
EF² = DG² + FG²
Substituting the given values, we get:
(n + 9)² = DG² + (4n - 6)²
Since vec DF bisects angle EDG, DG is equal to FG.
(n + 9)² = FG² + (4n - 6)²
Simplifying,
n² + 18n + 81 = FG² + 16n² - 48n + 36
Rearranging,
0 = 15n² - 66n - 45
Factoring,
0 = (3n + 5)(5n - 9)
Setting each factor equal to zero,
3n + 5 = 0 or 5n - 9 = 0
Solving for n,
3n = -5 or 5n = 9
n = -5/3 or n = 9/5
Since n represents a length, we can disregard the negative value.
Therefore, n = 9/5.
To find FG, we substitute the value of n in the equation FG = 4n - 6:
FG = 4(9/5) - 6
FG = 36/5 - 30/5
FG = 6/5
So, FG = 6/5.
EF=n+9
FG=4n-6
E=90 degrees
G=90 degrees
Enter your answer and show all the steps that you use to solve this problem in the space provided. vec DF bisects angle EDG Find FG. The diagram is not to scale
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