Question

已知四棱台上，下底面对应边分别是a，b，试求其中截面把此棱台侧面分成的两部分面积之比．

Answer 1

设A₁B₁C₁D₁是棱台ABCD-A₂B₂C₂D₂的中截面，延长各侧棱交于P点．
∵BC=a，B₂C₂=b∴B₁C₁=

a+b

2

∵BC∥B₁C₁∴

S△PBC

S△PB1C1

=

a2

( a+b 2 )2

∴S△PB1C1=

(a+b)2

4a2

•S△PBC
同理S△PB2C2=

b2

a2

•S△PBC
∴

SB1C1CB

SB2C2C1B1

=

S△PB1C1=S△PBC

S△PB2C2-S△PB1C1

=

(a+b)2 4a2 -1

b2 a2 - (a+b)2 4a2

=

b2+2ab-3a2

3b2-2ab-a2

=

(b+3a)(b-a)

(3b+a)(b-a)

=

b+3a

3b+a

同理：

SABB1A1

SA1B1B2A1

=

SDCC1D1

SD1C1C2D2

=

SADD1A1

SA1D1D2A1

=

b+3a

3b+a

由等比定理，得

S上棱台侧

S下棱台侧

=

3a+b

a+3b

故中截面把此棱台侧面分成的两部分面积之比为：

S上棱台侧

S下棱台侧

=

3a+b

a+3b

．