In the diagram, FH2FG, GHHI, and FIIK. Which of the following statements must be true?

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

(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