Question

3 - living room
find each missing side. then find the corresponding letters for each answer.

1. gh =
2. gi =
3. jk =
4. kl =

your lock code is your list of letters in order.
enter your lock code below using all capitals and no spaces. example: uwxs
your answer

Explanation:

Step1: Analyze triangle GHI

In right - triangle GHI with $\angle I = 60^{\circ}$ and $HI = 6$. We know that $\tan60^{\circ}=\frac{GH}{HI}$ and $\cos60^{\circ}=\frac{HI}{GI}$.
Since $\tan60^{\circ}=\sqrt{3}$ and $HI = 6$, then $GH=HI\times\tan60^{\circ}=6\sqrt{3}$. Since $\cos60^{\circ}=\frac{1}{2}$ and $HI = 6$, then $GI = 2\times HI=12$.

Step2: Analyze triangle JKL

In right - triangle JKL with $\angle J = 30^{\circ}$ and $JL = 16$. We know that $\sin30^{\circ}=\frac{KL}{JL}$ and $\cos30^{\circ}=\frac{JK}{JL}$.
Since $\sin30^{\circ}=\frac{1}{2}$ and $JL = 16$, then $KL=\frac{1}{2}\times JL = 8$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$ and $JL = 16$, then $JK = 8\sqrt{3}$.
Matching with the given options: $GH = 6\sqrt{3}$ (corresponds to W), $GI = 12$ (corresponds to Y), $JK = 8\sqrt{3}$ (not in the options for this part of the code - building), $KL = 8$ (not in the options for this part of the code - building). So the lock - code is WY.

Answer:

1. $6\sqrt{3}$
2. $12$
3. $8$
4. $8\sqrt{3}$

Lock - code: WY