Cho hình chóp S.ABCD, đáy ABCD là hình vuông cạnh a, ΔSAD đều và (SAD)(ABCD).H là trung điểm AD
a) Tính d(D,(SBC));d(HC,SD)
b) mp(α) qua H và vuông góc với SC tại I. Chứng tỏ (α) cắt các cạnh SB,SD và tính [B,SC,D]

Ta có SAAD
{SH(ABCD)SH=a32(ΔSABđu)
Chọn hệ trục tọa độ Hxyz sao cho: H(0;0;0),A(a2;0;0),S(0;0;a32)
D(a2;0;0),B(a2;a;0),C(a2;a;0)

a) Ta có:
n(SBC)=[SB,SC]=[(a2;a;a32);(a2;a;a32)]=a22(0;3;2)(SBC):3y+2za3=0d(D,(SBC))=a37d(HC,SD)=|[HC,SD].HS||HC,SD]|=|[(a2;a;0),(a2;0;a32)](0;0;a32)||[(a2;a;0),(a2;0;a32)]|=a319

b) Ta có: SC=a2(1;2;3) Phương trình (α):x2y+3z=0
Phương trình tham số SC:{x=ty=2tz=a32+3t
I(3a16;6a16;5a316)
SB=a2(1;2;3) Phương trình tham số SB:{x=ty=2tz=a323t
Gọi M=SB(α)M(a4;a2;a34)
xS<xM<xB nên MSB
SD=(a2;0;a32)=a2(1;0;3)
Phương trình tham số SD:{x=a2+ty=0z=3t
Gọi N=SD(α)N(3a8;0;a38)
xD<xN<xSNSB
Tính [B,SC,D]=φ
cosφ=cos^MIN=IM.INIM.IN=a16(7;2;3).(3a6).(1;2;3)a16.3a16(49+4+3)(1+4+3)=77
Vậy cosφ=77
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