Question

For the right triangles below, find the exact values of the side lengths c and h. If necessary, write your responses in simplified radical form. 0/6 C = . 300 h Х 5 ? 452 h = 2

Answer 1

Step-by-step explanation:

We apply sine rule to find the missing sides in each of the triangle.

Reason: Angles and one side are given

a/sin A = c/sinC

a = 4, A = 90°

c = ?, C = 45°

`\begin{gathered} (4)/(\sin90\degree)=(c)/(\sin45\degree) \\ \text{cross multiply:} \\ 4(\sin \text{ 45) = c (sin 90)} \\ \end{gathered}`
`\begin{gathered} In\text{ radical form: sin45 = }\frac{1}{\sqrt[]{2}}=\text{ }\frac{\sqrt[]{2}}{2} \\ 4(\frac{\sqrt[]{2}}{2})\text{ = c(}1) \\ 2\sqrt[]{2}\text{= c} \\ c\text{ = 2}\sqrt[]{2} \end{gathered}`

a/sinA = h/sinH

a = 2, A = 30°

h = ?, H = 60°

`\begin{gathered} (2)/(\sin30)=(h)/(\sin60) \\ In\text{ radical form:} \\ \sin \text{ 30 = 1/2, sin }60=\text{ }\frac{\sqrt[]{3}}{2} \\ (2)/((1)/(2))=\frac{h}{\frac{\sqrt[]{3}}{2}} \\ \end{gathered}`
`\begin{gathered} 2*(2)/(1)=h*\frac{2}{\sqrt[]{3}} \\ 4\text{ = }\frac{\text{2h}}{\sqrt[]{3}} \\ 4\sqrt[]{3}\text{ = 2h} \\ \frac{4\sqrt[]{3}}{2}=(2h)/(2) \\ h\text{ = }2\sqrt[]{3} \end{gathered}`