Answer:
x = 2
DG = 4
Step-by-step explanation:
The thing to know about the incenter of a triangle, is that the segments to each side are equidistant. In other words, they are identical lengths.
This means that ED = FD = GD.
Which means:
DE = DF
x^2 = 20 - 8x
So we need to solve for X. I would solve this using factoring:
Subtract 20 from both sides.
x^2−20=−8x
Add 8x to both sides.
x^2−20+8x=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
x^2+8x−20=0
Factor the left hand side by grouping. First, the left hand side needs to be rewritten as x^2+ax+bx−20. To find a and b, set up a system to be solved.
a+b=8 ab=1(−20)=−20
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product −20.
(−1,20) (−2,10) (−4,5)
Calculate the sum for each pair.
−1+20=19
−2+10=8
−4+5=1
The solution is the pair that gives sum 8.
a=−2 b=10
Rewrite x^2+8x−20 as (x^2−2x)+(10x−20).
(x^2−2x)+(10x−20)
Factor out x in the first and 10 in the second group.
x(x−2)+10(x−2)
Factor out common term x−2 by using distributive property.
(x−2)(x+10)
To find equation solutions, solve x−2=0 and x+10=0.
x = 2
x = −10
In this case, X is equal to 2 as the solution -10 doesn't make much sense given the context of the problem.
To find DG we just need to plug the solution back into one of the equations and solve:
20 - 8x
20 - 8(2)
20 - 16 = 4
DG = 4
