Question

find each missing side. then find the corresponding letters for each answer. 1) ac = __ 2) bc = 3) xz = 4) yz = __ your lock code is your list of letters in order. enter your lock code below using all capitals and no spaces. example: mkjp your answer

Explanation:

Step1: Analyze triangle ABC

In right - triangle ABC with $\angle B = 90^{\circ}$ and $\angle C=60^{\circ}$, $\angle A = 30^{\circ}$. The side opposite $\angle A$ is $BC$, the side opposite $\angle B$ is $AC$ (hypotenuse), and the side opposite $\angle C$ is $AB = 5\sqrt{3}$.
We know the trigonometric ratio $\sin C=\frac{AB}{AC}$ and $\cos C=\frac{BC}{AC}$. Also, in a 30 - 60 - 90 triangle, if the side opposite the $30^{\circ}$ angle is $x$, the side opposite the $60^{\circ}$ angle is $x\sqrt{3}$ and the hypotenuse is $2x$. Since $AB = 5\sqrt{3}$ (opposite $60^{\circ}$ angle), then $BC = 5$ (opposite $30^{\circ}$ angle) and $AC = 10\sqrt{3}$.

Step2: Analyze triangle XYZ

In right - triangle XYZ with $\angle Y = 90^{\circ}$ and $\angle X = 30^{\circ}$, if the side opposite $\angle X$ is $YZ = 15$, then the hypotenuse $XZ$ (opposite the right - angle) is given by $\sin X=\frac{YZ}{XZ}$. Since $\sin30^{\circ}=\frac{1}{2}$, and $YZ = 15$, then $XZ = 30$. Also, the side opposite the $60^{\circ}$ angle ($\angle Z$) is $YZ\sqrt{3}=15\sqrt{3}$.

Answer:

1. $AC = 10\sqrt{3}$, N
2. $BC = 5$, M
3. $XZ = 30$, L
4. $YZ = 15\sqrt{3}$, J

Your lock code: NM LJ