Question

The figure below shows △ ABC and it’s dilation image △ EFG

which statement is true?
CB = 2GF

∠A ≅ ∠G

CB = EF

GF = 2CB

Answer 1

`\mathrm{CB=2GF\ is\ correct.}`

Explanation:

`\mathrm{The\ dilation\ image\ \triangle\ EFG\ and\ it's\ object\ \triangle\ ABC\ are\ similar\ triangles.}\\\mathrm{We\ know\ that\ the\ corresponding\ sides\ of\ similar\ triangles\ are\ proportional}\\\mathrm{and\ their\ corresponding\ angles\ are\ equal.}\\\mathrm{We\ are\ given\ that\ AB=2EF,\ AC=2GE.\ So,\ BC\ is\ also\ equal\ to\ 2GF.}`

`\bold{Why\ is\ \angle A\ not\ congruent\ to\ \angle G?}`

`\rightarrow\ \mathrm{Since\ \triangle ABC\ and\ \triangle EFG\ are\ similar,}\\\mathrm{\angle A=\angle E,\ \angle C=\angle G\ and\ \angle B=\angle F\ because\ corresponding\ angles\ of}\\\mathrm{similar\ triangles\ are\ congruent(equal).}\\\mathrm{Let's\ assume\ that\ \angle A \cong\ \angle G.}\\\mathrm{This\ implies\ \angle A=\angle C\ because\ \angle C\ and\ \angle\ G\ are\ congruent.}\\\mathrm{This\ would\ mean\ that\ \triangle ABC\ is\ isosceles\ triangle\ as\ base\ angles\ are\ equal.}\\`

`\mathrm{And\ AB\ should\ be\ equal\ to\ BC\ since\ angles\ A\ and\ C\ are\ equal.\ So\ }\\\mathrm{AB=BC=30.}\\\mathrm{But\ we\ have\ a\ theorem\ that\ says\ sum\ of\ two\ sides\ of\ a\ triangle\ should\ be}\\\mathrm{greater\ than\ the\ third\ side.\ But\ in\ this\ case,\ AB+BC \ is\ not\ greater\ than}\\\mathrm{AC.\ Hence,\ \angle A\ and\ \angle G\ cannot\ be\ congruent.}`

`\bold{Why\ CB\ cannot\ be\ equal\ to\ EF?}`

`\rightarrow\ \mathrm{CB=EF\ would\ mean\ that\ CB=15.}\\\mathrm{So\ once\ again,\ in\ triangle\ ABC,\ AB+BC\ is\ not\ greater\ than\ AC.\ The}\\\mathrm{theorem\ stating\ sum\ of\ two\ sides\ in\ a\ triangle\ is\ greater\ than\ third\ side\ is}\\\mathrm{unsatisfied.\ Hence,\ CB\ and\ EF\ cannot\ be\ equal.}`

`\bold{Why\ GF \\e 2CB?}\\\rightarrow \mathrm{This\ clearly\ opposes\ the\ theorem\`

Answer 2

CB = 2GF

Explanation:

Because △ EFG is a dilation of △ ABC, they are similar triangles. Therefore, the ratios of their corresponding side lengths are equal:

AC=20=2(10)=2EG
AB=30=2(15)=2EF
CB=2GF

Also, ∠A and ∠G are not corresponding angles, so there's no proof of their congruence; CB and EF aren't corresponding sides; and GF is shorter than CB, so multiplying CB by 2 gives the wrong length for GF.