Answer:
6 units
Step-by-step explanation:
Given: Points H and F lie on circle with center C. EG = 12, EC = 9 and ∠GEC = 90°.
To find: Length of GH.
Sol: EC = CH = 9 (Radius of the same circle are equal)
Now, GC = GH + CH
GC = GH + 9
Now In ΔEGC, using pythagoras theorem,
[tex](Hypotenuse)^{2} = (Base)^{2} +(Altitude)^{2}[/tex] ......(ΔEGC is a right triangle)
[tex](GC)^{2} = (GE)^{2} +(EC)^{2}[/tex]
[tex](GH + 9)^{2} = (9)^{2} +(12)^{2}[/tex]
[tex](GH )^{2} + (9)^{2} + 18GH = 81 + 144[/tex]
[tex](GH )^{2} + 81 + 18GH = 81 + 144[/tex]
[tex](GH )^{2} + 18GH = 144[/tex]
Now, Let GH = x
[tex]x^{2} +18x = 144[/tex]
On rearranging,
[tex]x^{2} +18 x - 144 = 0[/tex]
[tex]x^{2} - 6x +24x + 144 = 0[/tex]
[tex]x (x-6) + 24 (x - 6) =0[/tex]
[tex](x - 6) (x + 24) = 0[/tex]
So x = 6 and x = - 24
∵ x cannot be - 24 as it will not satisfy the property of right triangle.
Therefore, the length of line segment GH = 6 units. so, Option (D) is the correct answer.
