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
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,
Putting AD= 40 cm and CD = 90 cm in the above relation:
`rArr BD=60` cm.
Therefore, the length of the pole BD is 60 cm and angle ABC=`90^o`.