Question

Can someone help me with this geometry question I’m confused also it’s a 2 part question

Answer 1

XZ = 6 units

YZ = 10.39 units

Step-by-step 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):

`\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=(opposite)/(hypotenuse) \\ sin\theta=(opposite)/(hypotenuse) \\ \text{Substituting the known variables into the equation, we have:} \\ sin30^(\circ)=(XY)/(12) \\ \text{Cross multiply to obtain length ''XY'', we have:} \\ XY=12* sin30^(\circ) \\ \text{From Mathematical tables, }sin30^(\circ)=0.5 \\ XY=12*0.5 \\ XY=6 \\ \\ \therefore XY=6units \end{gathered}`

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):

`\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=(adjacent)/(hypotenuse) \\ cos\theta=(adjacent)/(hypotenuse) \\ \text{Substituting the variables into the formula, we have:} \\ cos30^(\circ)=(YZ)/(12) \\ \text{Cross multiply to obtain ''YZ'', we have:} \\ YZ=12* cos30^(\circ) \\ \text{From Mathematical tables, }cos30^(\circ)=0.8660 \\ YZ=12*0.8660 \\ YZ=10.392\approx10.39 \\ YZ=10.39units \\ \\ \therefore YZ=10.39units \end{gathered}`

Method 2 (Pythagoras Theorem):

`\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}`

YZ = 10.39 units