# Points A,B and C are taken in the ascending order lie on a straight line inclined at an angle `theta` to the horizontal. AB = x and D is the point vertically above at height h from point C. CD...

Points A,B and C are taken in the ascending order lie on a straight line inclined at an angle `theta` to the horizontal. AB = x and D is the point vertically above at height h from point C. CD subtends angles `alpha` and `beta` at A and B respectively.

Please Prove that;

`h = (xsinalphasinbeta)/(sin(beta-alpha)costheta)`

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The situation is attached in the image.

According to the description given and using the image;

`angleDAB = alpha`

`angleDBC = beta`

`angleCAE = theta`

Applying sine law for triangle ACD

`h/(sinalpha) = (AD)/(sin(pi/2+theta))`

`h = (AD)/(sin(pi/2+theta))xxsinalpha`

Applying sine law for triangle ABD

`(AD)/(sin(pi-beta)) = x/(sin(beta-alpha))`

`AD = x/(sin(beta-alpha))xxsin(pi-beta) `

`AD = (xsinbeta)/(sin(beta-alpha))`

`h = (AD)/(sin(pi/2+theta))xxsinalpha`

`h = (xsinbeta)/(sin(beta-alpha))xx(sinalpha)/(costheta)`

*So the required answer is proved.*

`h = (xsinbetasinalpha)/(sin(beta-alpha)costheta)`