a Let's find out as much as possible about △RSL using the given information in the diagram. For starters, we see that ∠RSL and ∠TSR form a linear pair. By the Linear Pair Postulate, these angles are supplementary which means their measures sum to 180∘.
Let's add this piece of information to our diagram.
According to the Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is 180∘. Therefore, we can write the following equation:
The measure of ∠L is also 35∘ which means △RSL is an isosceles triangle. Using the Converse of the Base Angles Theorem, we can therefore say that RS≅SL.
Thus, the length of RS is also 2.1mi.
b The distance from the boat to the shoreline, d, is constant since the boat travels parallel to the shoreline. Below, we have outlined the position of the boat at three different locations.
At the original position, let's find the point on the shoreline where the angle from the boat is 45∘. At the same time, we also start measuring the distance traveled.
As the boat continues traveling, the horizontal distance (as seen from above) between the boat and P shrinks. When the boat is directly above P, whatever distance you have traveled from the original position, is also the distance to the shoreline. This is because the traveled distance and the distance to the shoreline, will form the legs of an isosceles right triangle as seen below.
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