Question

Help finding the missing side lengths for RQ and QP

Answer 1

We have the following triangle

We have the hypotenuse and all the angles so we can use the law of sines

`(a)/(\sin(A))=(b)/(\sin(B))=(c)/(\sin(C))`

In this case take us a = 4, b = QP andc = RQ

First, we solve QP

`\begin{gathered} (4)/(\sin(90))=(b)/(\sin(30)) \\ b=(4)/(\sin(90))\cdot\sin (30) \\ b=2 \end{gathered}`

Second. we solve RQ

`\begin{gathered} (4)/(\sin(90))=(c)/(\sin(60)) \\ c=(4)/(\sin(90))\cdot\sin (60) \\ c=2\sqrt[]{3} \end{gathered}`

These are the solutions

To check this we can take out the hypotenuse using the Pythagoras theorem and check that

`\begin{gathered} H=\sqrt[]{(2)^2+(2\sqrt[]{3})^2} \\ H=\sqrt[]{4+(4\cdot3)} \\ H=\sqrt[]{4+12} \\ H=\sqrt[]{16} \\ H=4 \end{gathered}`

This is correct

In conclusion, these answers are:

`\begin{gathered} QP=2 \\ RQ=2\sqrt[]{3} \end{gathered}`