如图,在梯形ABCD中,AD∥BC,AC⊥AB,AD=CD,cosB=5/13,BC=26. 求:(1)cos∠DAC的值; (2)线段AD的长.

问题描述:

如图,在梯形ABCD中,AD∥BC,AC⊥AB,AD=CD,cosB=

5
13
,BC=26.
求:(1)cos∠DAC的值;
(2)线段AD的长.

(1)在Rt△ABC中,∠BAC=90°,cosB=

AB
BC
5
13

∵BC=26,
∴AB=10.
∴AC=
BC2−AB2
262102
=24

∵AD∥BC,
∴∠DAC=∠ACB.
∴cos∠DAC=cos∠ACB=
AC
BC
12
13

(2)过点D作DE⊥AC,垂足为E,
∵AD=CD,AC=24,
∴AE=EC=
1
2
AC=12,又AD=DC,
∴在Rt△ADE中,cos∠DAE=
AE
AD
12
13

∴AD=13.