Geometry question involving triangles given with picture.

Question

Here's the question:

• $\overset{\Delta}{ABC}$ is a triangle.
• $D$ is a point on $[BC]$.
• $|BD|=4$.
• $|AD|=|CD|$.
• $\text m(\widehat{CBA})=\alpha=30^\circ$.
• $\text m(\widehat{ACB})=\beta=20^\circ$.
• What is $\color{magenta}{|AC|}=\color{magenta}x$?

Geometric approaches gave me nothing. Letting $|AD|=|CD|=a$, $|AB|=b$ and applying law of cosines to all three triangles yields;

$$ \begin{align} x&=2a\cos 20^\circ\\ 16&=a^2+b^2+2ab\cos 70^\circ\\ (a+4)^2&=b^2+x^2+2bx\cos 50^\circ \end{align} $$

respectively, from $\overset{\Delta}{ACD}$, $\overset{\Delta}{ABD}$ and $\overset{\Delta}{ABC}$. Directly isolating $x$ from these equations gives an expression involves trigonometric function(s), and i cannot find a way to annihilate trigonometric functions in these equations to reach the answer which is $\color{magenta}x=\color{magenta}4$.

Answer 1

Here is an elegant purely geometric solution. The diagram shows the 18-gon that I used to find the solution, but the solution itself does not need the 18-gon.

Let $E$ be the image of $B$ under the reflection that maps $A$ to $D$.

Then $\angle EAC = \angle EAD + \angle DAC = \angle ADB + \angle DAC = 40^\circ + 20^\circ = 60^\circ$.

Also $\angle DEA = \angle DBA = 30^\circ$.

Thus $DE$ bisects $AC$ because $DE \perp AC$ and $\triangle ACD$ is isosceles.

Thus $\triangle EAC$ is equilateral and hence $\overline{AC} = \overline{AE} = \overline{BD}$.

Answer 2

First calculate some angles:

$ \angle DAC = \angle DCA = 20^o $ (isosceles triangle theorem https://en.wikipedia.org/wiki/Isosceles_triangle )

$ \angle CAD = 180^o -20^o -20^o = 140^o $ (triangle is $180^o$)

$ \angle CAB = 180^o -30^o -20^o = 130^o $ (triangle is $180^o$)

$ \angle BAD = \angle BAC- \angle DAC = 130^o-20^o = 110^o $

and the rest is all the law of sinus ( https://en.wikipedia.org/wiki/Law_of_sines ):

$ \frac{AD}{\sin (\angle ABD)} = \frac{BD} {\sin(\angle BAD)} $

$ \frac{AD}{\sin (\angle ACD)} = \frac{AC} {\sin(\angle ADC)} $

done :)

Answer 3

Drop a perpendicular from $D$ to $M$ on $AC$. By congruence of the resultant $\bigtriangleup AMD$ and $\bigtriangleup MDC\,(*)$ we observe that $|AM| = x/2$ and so $|AD| = (x/2)\sec20^{\circ}$.

Now we note that $A\hat{D}B = 20\times2 = 40^{\circ}$ since the exterior angle of a triangle equals the sum of $2$ opposite interior angles.

So by the angle sum of $\bigtriangleup ABD,\,B\hat{A}D = 180-30-40 = 110^{\circ}$. Then, applying the sine rule: $$\frac{AD}{\sin A\hat{B}D} = \frac{4}{\sin B\hat{A}D}$$ we have $$ \frac{(x/2)\sec20^{\circ}}{\sin30^{\circ}} = \frac{4}{\sin 110^{\circ}}$$ which by rearrangement, yields $|AC|$:$$x = 8\sin 30^{\circ} \cos20^{\circ} \mathrm{cosec}\,110^{\circ} = 4$$

$(*)$ The congruence can be seen from the facts that:

• $|AD| = |DC|$
• $|MD|$ is a common length
• $A\hat{M}D = D\hat{M}C$ since $DM\perp AC$

satisfying the $RHS$ test for right triangles.

Answer 4

Construct isoceles $\triangle BDF$ with common angle $20^\circ$ and isoceles $\triangle BDG$ with common angle $30^\circ$. Now $\angle BGF=\angle FGD=\angle DGA=60^\circ$, and $\angle GDF=\angle ADG=10^\circ$. So $\triangle GDF\cong \triangle GDA$ and so $\overline{DF}=\overline{DA}$. Since $\triangle ADC\sim\triangle DFB$, we have the desired result.