In the diagram,triangle ABC is the cross section of a tent and BD,which represents the supporting pole,is perpendicular to the ground AC.It is given that triangle ABD is similar to triangle BCD,AD= 40 cm and CD = 90 cm.

Find the length of the pole BD and angle ABC.Thanks

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Given that `Delta ABD` is similar to `Delta BCD` . Hence the corresponding angles of the two triangles are equal i.e.

`angle ADB` =`angle BDC` = `90^o` (given)

`angle ABD =angle BCD`

`angle BAD =angle CBD`

From `Delta ABD` ,

`angle ABD + angle BAD =90^o`

`rArr angle ABD + angleCBD =90^o`

`rArr angle ABC =90^o`

When we drop an altitude (BD in the given image ) from the right angle ABC of a right triangle, the length of the altitude becomes a geometric mean. This occurs because you end up with similar triangles which have proportional sides and the altitude is the long leg of one triangle (`Delta ABD` ) and the short leg of the other similar triangle (`Delta BCD` ). Hence,

`(AD)/(BD)=(BD)/(CD)`

Putting AD= 40 cm and CD = 90 cm in the above relation:

`BD^2=40*90`

`rArr BD=60` cm.

**Therefore, the length of the pole BD is 60 cm and angle ABC=`90^o`**.

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