Answer:
Since GH is the midsegment of triangle AABC, it is parallel to side AB and its length is equal to half the length of AB.
Therefore, GH = AB/2.
In terms of x, AB = AC + CB = x + 22.
So GH = (x + 22)/2.
Since GH is also the length of the segment joining the midpoints of sides AC and CB, we can use the midpoint formula to find the coordinates of point H in the coordinate plane.
Let the coordinates of points A, B, and C be (0,0), (AB,0), and (x,GC), respectively.
Then the coordinates of point H are ((0+x)/2, (0+GC)/2) = (x/2, GC/2).
Since GH is perpendicular to AC, the product of their slopes is -1.
The slope of AC is (GC - 0)/(x - 0) = GC/x.
The slope of GH is (0 - GC/2)/(AB/2 - x/2) = -GC/(x+22).
Therefore, we have:
GC/x * (-GC/(x+22)) = -1
Simplifying this equation, we get:
GC^2 = x(x+22)
Substituting GH = GC/2 and GH = (x+22)/2, we get:
(GH)^2 = x(x+22)
Substituting GH = 11, we get:
11^2 = x(x+22)
121 = x^2 + 22x
x^2 + 22x - 121 = 0
Using the quadratic formula, we get:
x = (-22 +/- sqrt(22^2 - 41(-121)))/2
x = (-22 +/- 30)/2
x = 4 or x = -26
Since x represents a length, the only valid solution is x = 4.
Explanation: