Question

if ( bc = 90 mathrm{cm}, af = 29 mathrm{mm} ), and ( fc = 40 mathrm{mm} ), what would be the length of ( ef )? round your answer to two decimal places if necessary.

( ef=)

cm

Explanation:

Step1: Convert units

Since \(BC = 90\space cm\) and \(AF = 29\space mm=2.9\space cm\), \(FC = 40\space mm = 4\space cm\). Then \(AC=AF + FC=2.9+4 = 6.9\space cm\).

Step2: Use the similarity of triangles

Because \(\angle AEH=\angle AB D\) (given the equal - angle mark) and \(EH\parallel BD\) (since \(EH\) and \(BD\) are both perpendicular to \(AH\) in the isosceles - like sub - triangles), \(\triangle AEF\sim\triangle ABC\).
By the property of similar triangles \(\frac{EF}{BC}=\frac{AF}{AC}\).
Substitute \(BC = 90\space cm\), \(AF = 2.9\space cm\), \(AC = 6.9\space cm\) into the formula: \(EF=\frac{AF\times BC}{AC}\).

Step3: Calculate the value of \(EF\)

\(EF=\frac{2.9\times90}{6.9}=\frac{261}{6.9}\approx37.83\space cm\)

Answer:

\(37.83\space cm\)