We want to know how long it takes to travel across the swamp from point Z to point Y. Let's take a look at the given picture.
Let's begin by finding the length of side YZ. Then we can use the information from the exercise to calculate the time it takes for a swamp vehicle to travel across the swamp. First, notice that ∠ W is congruent to ∠ Z — they are both right angles, so their measures are 90^(∘).
∠ W = ∠ Z = 90^(∘)
We can also see that ∠ YXZ and ∠ VXW are vertical angles. We know that vertical angles are congruent, so let's mark it on our graph.
We found that two angles in △ XWV are congruent to two angles in △ XZY. Because of this, the third angles are also congruent. This means that the triangles are similar! Now we can use the fact that corresponding sides in similar triangles have equivalent ratios. Let's take a look at what it means in our case. We can mark the unknown distance from point Z to Y as x.
WV/WX=ZY/XZ ⇕ 10/6=x/3
Let's solve the equation for x.
We found that distance from point Z to point Y is 5 miles. From the exercise, we know that the swamp vehicle travels at 3.2 miles per hour. Let's use this information to create a proportion to find how long it takes for the swamp vehicle to travel across the swamp. We will mark the time needed to travel across the swamp as y.
3.2/1=5/y
Let's solve this equation for y!
