Weird angle! ... or is is?

Level 2

$\triangle ABC$ is acute with $\angle B = 2\angle C$ . $AD$ is the internal bisector of $\angle A$ such that $AB = DC$ . Find (in degrees) the measure of $\angle BAC$ .

NOTE -I found it in a mathematical contest. I solved it but am looking for a better solution and I'm sure some of you will come up with amazing ones.

The answer is 72.

1 solution

Shamik Banerjee
Jan 19, 2014

Say, angle C = α. It implies angle B = 2α, angle BAC = 180 – 3α, angle BAD = angle DAC = 90 – 3α/2 angle BDA = 90 – α/2, angle CDA = 90 + α/2.

Using sine rule in ABD, we get: AB/sin(90 – α/2) = AD/sin(2α) ==> AB/AD = {cos(α/2)}/{sin(2α)} … (1) Using sine rule in ADC, we get: DC/sin(90 – 3α/2) = AD/sin(α) ==> DC/AD = {cos(3α/2)}/{sin(α)} … (2) As AB = DC, from equations (1) & (2), we get {cos(α/2)}/{sin(2α)} = {cos(3α/2)}/{sin(α)} ==> sin(α) * cos(α/2) = sin(2α) * cos(3α/2) ==> (1/2) * {sin(3α/2) – sin(α/2)} = (1/2) * {sin(7α/2) – sin(α/2)} ==> sin(3α/2) = sin(7α/2)

angle C > 0 ==> α > 0 angle B + angle C = 3α < 180 ==> α < 60

As sin(y) = sin(180 – y), we can solve sin(3α/2) = sin(7α/2) as (3α/2) + (7α/2) = 180 ==> 5α = 180 ==> α = 36 ==> angle BAC = 180 – 3 * 36 = 72

Well i got a more funnier solution. Not sure if its correct or not :P

Sagnik Saha - 7 years, 4 months ago

LEMMA 1 = in a triangle ABC if angle B = 2 angle C than b^2 = c(c+a)

since AD is angle bisector thus a-c/c = c/b thus c^2 = ab - bc and thus b^2 = ab + ac - bc thus b(b+c) = a(b+c) thus a=b and hence we get angle A = angle B = 2 angle C thus angle A = 72

aditya dokhale - 7 years, 4 months ago

Well what about this solution. We take $\angle B = 2 \theta$ and $\angle A = \theta$ . Now we take a point $P$ on BC such that $\triangle ABP$ is isosceles. We assume that $P$ is different from $D$ . But then we have $\angle APB = 2 \theta$ which implies $\angle PAC = \theta$ and thus $PA = PC$ . But we have $PA = AB$ which means $AB = PC$ . But its given that $AB = DC$ and thus $D,P$ must be the same point. From here it follows easily that $\angle BAC = 2 \theta$ and we solve for $\theta$ .

Sagnik Saha - 7 years, 4 months ago

Weird angle! ... or is is?

The answer is 72.

1 solution

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