Question

Worksheet 8.4 Solving Right Triangles Solve the Triangles. Round to the nearest tenth. 1) B 10 7 C 2 A

Answer 1

Solving the triangle involves finding the measures of all the sides and angles of the triangle. Since it is a right triangle, then you can use the Pythagorean theorem to find the measure of the missing side, that is, the measure of side c.

The formula of the Pythagorean Theorem is

`\begin{gathered} a^2+b^2=c^2 \\ \text{ Where c is the hypotenuse and} \\ a,b\text{ are the sides of the right triangle} \end{gathered}`

So, in this case, you have

`\begin{gathered} a=7 \\ b=2 \\ c=\text{?} \\ a^2+b^2=c^2 \\ \text{ Replacing} \\ (7)^2+(2)^2=c^2 \\ 49+4=c^2 \\ 53=c^2 \\ \text{ Apply square root to both sides of the equation} \\ \sqrt[]{53}=\sqrt[]{c^2} \\ 7.3=c \end{gathered}`

Now, to find the measure of angle B you can use the trigonometric ratio

`\tan (x)=\frac{\text{opposite side}}{\text{adjacent side}}`

Graphically,

So, you have

`\begin{gathered} \tan (B)=(2)/(7) \\ \text{ Apply the inverse function }\tan ^(-1)(x)\text{ to both sides of the equation} \\ \tan ^(-1)(\tan (B))=\tan ^(-1)((2)/(7)) \\ B=15.9\text{\degree} \end{gathered}`

Finally, to find the measure of angle A you can use the triangle sum theorem, which says that, the sum of the three interior angles in a triangle is always 180°. So, you have

`\begin{gathered} m\angle A+m\angle B+m\angle C=180\text{\degree} \\ m\angle A+15.9\text{\degree}+90\text{\degree}=180\text{\degree} \\ m\angle A+105.9\text{\degree}=180\text{\degree} \\ \text{ Subtract 105.9\degree from both sides of the equation} \\ m\angle A+105.9\text{\degree}-105.9\text{\degree}=180\text{\degree}-105.9\text{\degree} \\ m\angle A=74.1\text{\degree} \end{gathered}`

Therefore, the measures of all sides and angles after solving the triangle are

`\begin{gathered} A=74.1\text{\degree} \\ B=15.9\text{\degree} \\ C=90\text{\degree} \\ a=7 \\ b=2 \\ c=7.3 \end{gathered}`