Question

if ( bc = 85 mathrm{cm}, af = 25 mathrm{mm} ), and ( fc = 40 mathrm{mm} ), what would be the length of ( ef )? round your answer to two decimal places if necessary. ( ef = )____( mathrm{cm} )

Explanation:

Step1: Convert units

Since \(1\ cm = 10\ mm\), then \(AF = 25\ mm=2.5\ cm\) and \(FC = 40\ mm = 4\ cm\), so \(AC=AF + FC=2.5+4 = 6.5\ cm\).

Step2: Prove similarity

Because \(\angle AEF=\angle ABC\) (given) and \(\angle A=\angle A\) (common angle), so \(\triangle AEF\sim\triangle ABC\) (by AA similarity criterion).

Step3: Use similarity ratio

For similar triangles \(\triangle AEF\) and \(\triangle ABC\), the ratio of corresponding sides is equal. That is \(\frac{EF}{BC}=\frac{AF}{AC}\).
Substitute \(BC = 85\ cm\), \(AF = 2.5\ cm\), \(AC = 6.5\ cm\) into the formula: \(EF=\frac{AF\times BC}{AC}\).
\(EF=\frac{2.5\times85}{6.5}=\frac{212.5}{6.5}\approx32.69\ cm\)

Answer:

\(32.69\)