Angling all the way...

Geometry Level 4

Triangle ABC has AC = BC , angle ACB = 96 , D is a point in ABC such that angle DAB = 18 and angle DBA = 30 . What is the measure of angle ACD .

All the angles are in Degrees.

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The answer is 78.

Harshvardhan Mehta by Harshvardhan Mehta , 23, India

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3 solutions

Sammit Jain
Jun 26, 2015

After constructing the figure as said, I tried joining C to the midpoint of AB, lets call it M. Now, AM = AB/2 = ACsin(half of Angle C) = ACsin48 Also, using sine rule, AD/sin30 = AB/sin132 = AB/sin48 Therefore, AD = AC, giving us ACD = 78 degrees.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Up voted. Very good out of box approach. I solved by common path, Sin Law.

Niranjan Khanderia - 2 years, 11 months ago
Ahmad Saad
Apr 21, 2016

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Let A C = B C = 1 AC=BC=1 .

By law of cosines, we have

x 2 = 1 2 + 1 2 − 2 ( 1 ) ( 1 ) ( cos ⁡ 96 ) x^2=1^2+1^2-2(1)(1)(\cos~96) ⟹ \implies x = 1.4863 x=1.4863

By law of sines, we have

y sin ⁡ 30 = 1.4863 sin ⁡ 132 \dfrac{y}{\sin~30}=\dfrac{1.4863}{\sin~132} ⟹ \implies y = 1 y=1

Therefore △ C A D \triangle CAD is isosceles with A C = A D = 1 AC=AD=1 . So

θ = 180 − 24 2 = \theta=\dfrac{180-24}{2}= 78 \boxed{78}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Good way of thinking.

Niranjan Khanderia - 2 years, 11 months ago

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