Дано:
A B C D − ABCD - A B C D −
A D = 48 AD=48 A D = 48
B C = 16 BC=16 B C = 16
C F : D F = 5 : 3 CF:DF =5:3 C F : D F = 5 : 3
Найти:
E F − ? EF - ? E F − ?
Решение:
Пусть C F = 5 x CF = 5x C F = 5 x D F = 3 x DF = 3x D F = 3 x
Проведем ВН ∣ ∣ C D ВН||CD ВН ∣∣ C D
A H = A D − B C = 48 − 16 = 32 AH = AD - BC = 48 - 16 = 32 A H = A D − B C = 48 − 16 = 32
B H = 5 x + 3 x = 8 x BH = 5x + 3x = 8x B H = 5 x + 3 x = 8 x
△ E B K \triangle EBK △ E B K △ A B H \triangle ABH △ A B H
∠ B \angle B ∠ B − - −
∠ B A H , ∠ B E K \angle BAH, \angle BEK ∠ B A H , ∠ B E K − - −
⇒ △ E B K ∼ △ A B H \Rightarrow \triangle EBK \sim \triangle ABH ⇒ △ E B K ∼ △ A B H E K A H = B K B H \cfrac{EK}{AH} = \cfrac{BK}{BH} A H E K = B H B K E K 32 = 5 x 8 x \cfrac{EK}{32} = \cfrac{5x}{8x} 32 E K = 8 x 5 x E K = 32 ⋅ 5 8 = 20 EK = \cfrac{32 \cdot 5}{8} = 20 E K = 8 32 ⋅ 5 = 20 E F = E K + K F = 20 + 16 = 36 EF = EK + KF = 20 + 16 = 36 E F = E K + K F = 20 + 16 = 36
36 36 36
