Answer:
XZ = 6 units
YZ = 10.39 units
Explanation:
We were given the following information:
The figure is a right triangle having one known side and three known angles (90 degrees, 30 degrees & 60 degrees)
XZ = 12
XY = ?
YZ = ?
Part A
We are to calculate the length of side XY. This is shown below:
Since we have one known side and three known angles, we can obtain the length of XY using the Trigonometric Ratio (SOHCAHTOA):
[tex]\begin{gathered} \angle Z=30^{\circ} \\ \text{ In relation to angle Z, XZ is the hypotenuse \& XY is the opposite side:} \\ XZ=hypotenuse=12units \\ XY=opposite=\text{?} \\ \text{We will obtain ''XY'' using '}SOH^{\prime} \\ SOH\Rightarrow sin\theta=\frac{opposite}{hypotenuse} \\ sin\theta=\frac{opposite}{hypotenuse} \\ \text{Substituting the known variables into the equation, we have:} \\ sin30^{\circ}=\frac{XY}{12} \\ \text{Cross multiply to obtain length ''XY'', we have:} \\ XY=12\times sin30^{\circ} \\ \text{From Mathematical tables, }sin30^{\circ}=0.5 \\ XY=12\times0.5 \\ XY=6 \\ \\ \therefore XY=6units \end{gathered}[/tex]XY = 6 units
Part B
We are to calculate the length of side YZ. Since this is a right triangle, we can solve it using 2 different methods as shown below:
Method 1 (Trigonometric Ratio):
[tex]\begin{gathered} \angle Z=30^{\circ} \\ \text{In relation to angle Z, ''YZ'' is the hypotenuse and ''XY'' is the adjacent side:} \\ XZ=hypotenuse=12units \\ YZ=adjacent=\text{?} \\ \text{We will obtain ''YZ'' using '}CAH^{\prime} \\ CAH\Rightarrow cos\theta=\frac{adjacent}{hypotenuse} \\ cos\theta=\frac{adjacent}{hypotenuse} \\ \text{Substituting the variables into the formula, we have:} \\ cos30^{\circ}=\frac{YZ}{12} \\ \text{Cross multiply to obtain ''YZ'', we have:} \\ YZ=12\times cos30^{\circ} \\ \text{From Mathematical tables, }cos30^{\circ}=0.8660 \\ YZ=12\times0.8660 \\ YZ=10.392\approx10.39 \\ YZ=10.39units \\ \\ \therefore YZ=10.39units \end{gathered}[/tex]Method 2 (Pythagoras Theorem):
[tex]\begin{gathered} \text{Pythagoras Theorem is given by:} \\ c^2=a^2+b^2 \\ where\colon \\ c=hypotenuse=XZ=12 \\ a=side_1=YZ=\text{?} \\ b=side_2=XY=6 \\ \text{Substituting the variables into the formula, we have:} \\ 12^2=a^2+6^2 \\ 144=a^2+36 \\ \text{Subtract ''36'' from both sides, we have:} \\ 144-36=a^2 \\ 108=a^2 \\ a^2=108 \\ \text{Take the square root of both sides, we have:} \\ a=\sqrt[]{108} \\ a=10.392\approx10.39 \\ a=10.39 \\ a=YZ\Rightarrow YZ=10.39 \\ YZ=10.39units \\ \\ \therefore YZ=10.39units \end{gathered}[/tex]YZ = 10.39 units
