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DF−→−
bisects ∠EDG
. Find FG. The diagram is not to scale.
A diagram of two triangles drawn in the interior of an angle is shown. Angle upper E upper D upper G is shown with point upper F located in the interior of the angle. Point upper F is connected to point upper D with a line segment. Angle upper E upper D upper F and angle upper G upper D upper F both are lab
To find FG, we need to use the fact that DF bisects angle EDG. This means that angle EDF is equal to angle GDF.
Let's call the measure of angle EDF x. Then the measure of angle GDF is also x because they are equal.
We know that the sum of angles in a triangle is 180 degrees. So, angle EDG is equal to 180 - x - x = 180 - 2x.
Since DF bisects angle EDG, angle EDF + angle GDF = angle EDG. This can be written as x + x = 180 - 2x.
Combining like terms, we get 2x = 180 - 2x.
Adding 2x to both sides, we get 4x = 180.
Dividing both sides by 4, we get x = 45.
Now that we know the measure of angle EDF is 45 degrees, we can find angle EFG. Angle EFG is equal to 180 - angle EDF - angle DFG.
Substituting the values we know, angle EFG = 180 - 45 - 45 = 90 degrees.
Since angle EFG is a right angle, triangle EFG is a right triangle. Therefore, FG is the hypotenuse of this right triangle.
Using the Pythagorean theorem, we can find the length of FG. Let's call the length of EF a, and the length of EG b.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, FG^2 = EF^2 + EG^2.
Substituting the values we know, FG^2 = a^2 + b^2.
Since EF = a and EG = b, we can write FG^2 = (EF)^2 + (EG)^2.
Substituting the values we know, FG^2 = (ED - DF)^2 + EG^2.
Since DF = FG (because DF bisects angle EDG), we can simplify the equation to FG^2 = (ED - FG)^2 + EG^2.
Expanding the equation, we get FG^2 = ED^2 - 2ED * FG + FG^2 + EG^2.
Simplifying further, we get FG^2 - FG^2 = ED^2 - 2ED * FG + EG^2.
The FG^2 terms cancel out, leaving us with 0 = ED^2 - 2ED * FG + EG^2.
Rearranging the equation, we get 2ED * FG = ED^2 + EG^2.
Dividing both sides by 2ED, we get FG = (ED^2 + EG^2) / (2ED).
Now we can substitute the values we know: ED = 10 and EG = 8.
Calculating, we get FG = (10^2 + 8^2) / (2 * 10) = (100 + 64) / 20 = 164 / 20 = 8.2.
Therefore, FG is equal to 8.2.