Question:
In the diagram, FH = 2FG, GH = HI, and FI = IK. Which of the following statements must be true? Select all that apply.
(a) FG = HI
(b) HI = IJ
(c) IK = 3FG
(d) FH = GI
(e) HJ = JK
(f) HK = 2GI
Answer:
(a) FG = HI
(c) IK = 3FG
(d) FH = GI
(f) HK = 2GI
Explanation:
The diagram to this question has been attached to this response.
From the question:
=> FH = 2FG
Let FG be x units
Then FH = 2x units -----------------------(*)
=> FH = FG + GH [Remember: FG = x units, FH = 2x units]
2x = x + GH
GH = x units
Also
=> GH = HI [Remember: GH = x units]
x = HI
HI = x units
=> FG = GH = HI = x units --------------------(i)
=> FI = FH + HI [Remember: FH = 2x units, HI = x units]
FI = 2x + x
FI = 3x units
=> FI = IK [Remember: FI = 3x units ]
IK = 3x units -----------------------------(ii)
Now lets go to the question
(a) Check if FG = HI
FG = HI = x units [from equation (i)]
∴ FG = HI
(b) Check if HI = IJ
HI = x units [from equation (i)]
IJ = y units [Not enough information to find the IJ]
∴ HI = x units may not be equal to IJ
(c) Check if IK = 3FG
IK = 3x units [from equation (ii)]
FG = x units [from equation (i)]
∴ IK = 3FG
(d) Check if FH = GI
FH = 2x units [from equation (*)]
GI = GH + HI [GH = x units, HI = x units from equation (i)]
GI = x + x = 2x units
∴ FH = GI
(e) Check if HJ = JK
HJ = HI + IJ
HJ = (x + y) units
JK = IK - IJ
JK = 3x - y
∴ HJ may not be equal to JK
(f) Check if HK = 2GI
HK = HI + IJ + JK [JK = (3x - y) as calculated in (e) above]
HK = x + y + (3x - y)
HK = 4x
2GI = GI + GI [GI = 2x units as calculated in (d) above]
2GI = 2x + 2x = 4x
∴ HK = 2GI
