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Triangles

Long Answer Type

221. In the given Fig, O is a point in the interior of a triangle ABC, OD ⊥ BC,OE ⊥ AC and OF ⊥ AB. Show that
(i) OA² + OB² + OC² - OD² - OE² - OF² = AF² + BD² + CE^2,(ii) AF² + BD² + CE² = AE² + CD² + BF².

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Short Answer Type

222.

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

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223. A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

In right triangle ABC, we have
AC² = AB² + BC²
[By Pythagoras theorem]
$rightwards double arrow space space space space space space space space space space space space left parenthesis 24 right parenthesis squared space equals space left parenthesis 18 right parenthesis squared plus BC squared rightwards double arrow space space space space space space space space space space space space space 576 space equals space 324 plus BC squared rightwards double arrow space space space space space space space space space space space space space space BC squared space equals space 576 minus 324 rightwards double arrow space space space space space space space space space space space space space space BC squared space equals space 252 rightwards double arrow space space space space space space space space space space space space space space BC space equals space square root of 252 rightwards double arrow space space space space space space space space space space space space space space BC space equals space 6 square root of 7 space space straight m$

In right triangle ABC, we haveAC2 = AB2 + BC2[By Pythagoras theorem

Hence, the stake should be driven $6 square root of 7 space straight m$ for from the base of the pole so that the wire will be taut.

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Long Answer Type

224. An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after

1 1 half hours ?

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225.

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

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