For any $△ A B C,$ let the lengths of the sides opposite angles $A,$ $B,$ and $C$ be $a,$ $b,$ and $c,$ respectively.

The Law of Sines relates the sine of each angle to the length of the opposite side.

\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}

Let's start by finding

m ∠ F .

Then, we will use above law to find the values of

f

and

h .

We will find them one at a time.

Consider the triangle that satisfies the given conditions.

From the Triangle Angle Sum Theorem we know that the sum of the angles in a triangle is equal to

18 0^{\circ} .

With this information we can find

F .

m ∠ F + 4 0^{\circ} + 8 0^{\circ} = 18 0^{\circ} ⇕ m ∠ F = 6 0^{\circ}

Let's mark the measure of the angle $F$ on the graph.

We know that the length of a side is

14

and that the measure of its opposite angle is

8 0^{\circ} .

We also know that the measure of the angle that is opposite to the side we want to find is

6 0^{\circ} .

With this information and using the Law of Sines, we can write an equation in terms of

f .

\frac{sin 8 0 ^{\circ}}{1 4} = \frac{sin 6 0 ^{\circ}}{f}

Let's solve the above equation for

f

\frac{sin 8 0 ^{\circ}}{1 4} = \frac{sin 6 0 ^{\circ}}{f}

Cross multiply

sin 8 0^{\circ} \cdot f = sin 6 0^{\circ} \cdot 14

$LHS / sin 8 0^{\circ} = RHS / sin 8 0^{\circ}$

f = \frac{sin 6 0 ^{\circ} \cdot 1 4}{sin 8 0 ^{\circ}}

Use a calculator

f = 12.311393 \dots

Round to $1$ decimal place(s)

f \approx 12.3

Consider the triangle with the new information.

Exercise 7 Page 818

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