Question

Please solve it
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Answer 1

$\huge{\boxed{\mathcal\pink{\fcolorbox{red}{yellow}{Answer}}}}$

To prove:

The height of the point of interaction pf the lime joining the top of each tree to the foot of the opposite tree is given as ${\bold{\boxed{\red{\frac{ab}{a + b} metre}}}}$

solution:

height of the two trees AB and CD be b and a respectively.

Distance between the trees BC = p

Let EF = h be the height of point of intersection of the lines joining the top of each tree to the foot of the opposite trees.

Let CF = x then BF = (p – x)

In Δ’s ABC and EFC, ∠ABC = ∠EFC = 90° ∠C = ∠C (common angle)

$\frac{b}{p} = \frac{h}{x}$

${\bold{\boxed{\pink{\tt{bx=ph}}}}}$

∴

$x = \frac{ph}{b}$

ΔABC ~ ΔEFC (By AA similarity theorem) Similarly, ΔDCB ~ ΔEFB (By AA similarity theorem)

$\frac{a}{p} = \frac{h}{p - x } \\ ap - ax = ph$

$ap - a \times \frac{ph}{b} = ph$

$p(a - \frac{ah}{b} )= ph$

$\frac{ab-ah}{b} = h$

$ab -ah=bh$

$ab =ah+bh$

$ab= h(a+b)$

${\bold{\boxed{\red{h=\frac{ab}{a+b}}}}}$