Points H and F lie on circle c What is the length of line segment GH?
Accepted Solution
A:
Answer:6 unitsStep-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 + 9Now 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.