נתונים ( ∙ {\displaystyle \bullet } ) : ∠ A C B = 90 ∘ C D ⊥ A B ∠ A C E = ∠ B C E = 90 2 = 45 ∘ B C = 15 c m ∠ C A B = 25 ∘ {\displaystyle {\begin{aligned}&\angle ACB=90^{\circ }\\&CD\perp AB\\&\angle ACE=\angle BCE={\frac {90}{2}}=45^{\circ }\\&BC=15_{cm}\\&\angle CAB=25^{\circ }\\\end{aligned}}}
צ"ל : D E = ? {\displaystyle DE=?}
הוכחה : △ A B C ∠ B A C = 25 ∘ ∙ ∠ A C B = 90 ∘ ∙ ↓ ∠ A B C = 90 ∘ − 25 ∘ = 35 ∘ t r i a n g l e D C B ∠ C D B = 90 ∘ ∙ ↓ ∠ D C B = 90 ∘ − 65 ∘ = 25 ∘ ( ∗ 1 ) ∠ B C E = 45 ∙ ↓ ∠ D C E = ∠ B D E − ∠ B A C = 20 ∘ ( ∗ 2 ) △ B C D B C = 15 ∙ ∠ D C B = 25 ∘ ( ∗ 1 ) cos ∠ B C D = D C B C cos 25 = D C 15 ↓ D C = 13.595 △ D E C t a n ∠ D C E = D E C D t a n 20 = D E 13.595 D E = 4.948 ◼ {\displaystyle {\begin{aligned}&\triangle ABC\\&\angle BAC=25^{\circ }\bullet \\&\angle ACB=90^{\circ }\bullet \\&\downarrow \\&\angle ABC=90^{\circ }-25^{\circ }=35^{\circ }\\&triangleDCB\\&\angle CDB=90^{\circ }\bullet \\&\downarrow \\&\angle DCB=90^{\circ }-65^{\circ }=25^{\circ }(*1)\\&\angle BCE=45\bullet \\&\downarrow \\&\angle DCE=\angle BDE-\angle BAC=20^{\circ }(*2)\\&\triangle BCD\\&BC=15\bullet \\&\angle DCB=25^{\circ }(*1)\\&\cos \angle BCD={\frac {DC}{BC}}\\&\cos 25={\frac {DC}{15}}\\&\downarrow \\&DC=13.595\\&\triangle DEC\\&tan\angle DCE={\frac {DE}{CD}}\\&tan20={\frac {DE}{13.595}}\\&DE=4.948\blacksquare \\\end{aligned}}}
