נתון ( ∙ {\displaystyle \bullet } ): א. ∠ B E D = ∠ B A C = 90 ∘ B E = E C A D = 2 c m ∠ B C A = 36 ∘ {\displaystyle {\begin{aligned}&\angle BED=\angle BAC=90^{\circ }\\&BE=EC\\&AD=2_{cm}\\&\angle BCA=36^{\circ }\\\end{aligned}}}
צ"ל : B C = ? {\displaystyle \ BC=?}
הוכחה : E C = B E ∙ ∠ D E B = ∠ D E C = 90 ∘ D E = D E ↓ △ D E B ≅ △ D E C ∠ E B D = ∠ E C D = 36 ∘ △ A B C ∠ B A C = 90 ∘ ∙ ∠ B C A = 36 ∘ ∙ ∠ A B C = 90 ∘ − 36 ∘ = 54 ∘ ∠ A B D = ∠ A B C − ∠ D B C = 54 ∘ − 36 ∘ = 18 ∘ △ A B D tan ∠ A B D = A D A B tan 18 ∘ = 2 A B A B = 6.155 △ A B C cos ∠ A B C = A B B C cos 54 ∘ = 6.155 B C B C = 10.471 {\displaystyle {\begin{aligned}&EC=BE\bullet \\&\angle DEB=\angle DEC=90^{\circ }\\&DE=DE\\&\downarrow \\&\triangle DEB\cong \triangle DEC\\&\angle EBD=\angle ECD=36^{\circ }\\&\triangle ABC\\&\angle BAC=90^{\circ }\bullet \\&\angle BCA=36^{\circ }\bullet \\&\angle ABC=90^{\circ }-36^{\circ }=54^{\circ }\\&\angle ABD=\angle ABC-\angle DBC=54^{\circ }-36^{\circ }=18^{\circ }\\&\triangle ABD\\&\tan \angle ABD={\frac {AD}{AB}}\\&\tan 18^{\circ }={\frac {2}{AB}}\\&AB=6.155\\&\triangle ABC\\&\cos \angle ABC={\frac {AB}{BC}}\\&\cos 54^{\circ }={\frac {6.155}{BC}}\\&BC=10.471\\\end{aligned}}}
