Answer
$w = 6.7$
$x = 8.1$
Work Step by Step
Since we have the measure of an angle and the measure of the opposite side, we can use the tangent ratio to find $w$, which is the side adjacent to the given angle.
The tangent ratio is given as follows:
tan $A = \frac{opposite}{adjacent}$
Let's plug in what we know:
tan $56^{\circ} = \frac{10}{w}$
Multiply each side by $w$:
$w(tan 56^{\circ}) = 10$
Divide each side by tan $56^{\circ}$:
$w = 10$/tan $56^{\circ}$
Divide to solve:
$w = 6.7$
We have a bigger right triangle where we are given the measure of the angle and the side opposite. We are asked to find a segment of the adjacent side marked $x$. The adjacent side for this angle is made up of $w$, which we already found, and $x$, which we need to find. Let's use the tangent ratio again:
tan $A = \frac{opposite}{adjacent}$
Let's plug in what we know:
tan $A = \frac{10}{6.7 + x}$
Multiply each side by $6.7 + x$
$(tan 34^{\circ}){6.7 + x} = 10$
Divide both sides by tan $34^{\circ}$:
$6.7 + x = 10$/tan $34^{\circ}$
Subtract $6.7$ from each side of the equation to isolate $x$:
$x = 10$/tan $34^{\circ} - 6.7$
Do the division first:
$x = 14.8 - 6.7$
Subtract to solve for $x$:
$x = 8.1$
