A trivial triangle makeover

Geometry Level 4

A $\triangle ABC$ is isosceles ( $AB = AC$ ) with $\angle BAC = 120°$ . What would be $\angle A_{1}BC$ (in degrees) if distance between point $A_{1}$ and segment $BC$ is $h + a(1 + \sqrt{3})$ .

Notes :

$h$ - distance between segment $BC$ and point $A$

$a$ - length of $AB$ (or $AC$ )

$\triangle A_{1}BC$ is also isosceles ( $A_{1}B = A_{1}C$ )

The answer is 75.

1 solution

Milan Milanic
Jan 25, 2016

Solution:

$\triangle AEB$ (image above) is isosceles with $AB = AE = a$ and a side $BE$ labelled as $b = a\sqrt{3}$ ( sine law ).

$\triangle BEF$ is also isosceles with $BE = FE = b$ .

Now, a triangle $BFC$ is isosceles with $\angle BFC = 30°$ and it can be spotted that distance between point $F$ and segment $BC$ is $h + a(1 + \sqrt{3})$ . Therefore, point $F$ has all properties of point $A_{1}$ . Solution is $\boxed{75°}$ .

A trivial triangle makeover

The answer is 75.

1 solution

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