A tree standing on a horizontal plane is leaning towards east. At two points situated at distances $a$ and $b$ exactly due West of it, the angles of elevation of the top are respectively $\alpha$ and $\beta$. then the height of the top from the ground is $\dfrac { \left( b-a \right) \tan { \alpha } \tan { \beta } }{ \tan { \alpha } -\tan { \beta } }$.

Let the vertical height of the top of the tree be h and the horizontal distance it leans east of its base be x. From the point at distance a, we get the equation h equals (a + x) tan alpha, and from the point at distance b, we get h equals (b + x) tan beta. Equating the two expressions gives (a + x) tan alpha equals (b + x) tan beta, which simplifies to x equals (b tan beta minus a tan alpha) divided by (tan alpha minus tan beta); substituting this x back into the formula h equals (a + x) tan alpha eventually simplifies exactly to h equals ((b minus a) tan alpha tan beta) divided by (tan alpha minus tan beta), proving the statement is true.

• Thanks
• Wrong explanation