Question

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flight two planes take off from the same airstrip. the first plane flies west for 150 miles and then flies 30° south of west for 220 miles. the second plane flies east for 220 miles and then flies x° south of east for 150 miles. if x < 30, which plane is farther from the airstrip after the second leg? justify your answer.
the first plane; the legs are congruent. if x < 30, then the measure of the included angle, (180 − x)°, is greater for the second plane; so, by the hinge theorem, the second plane is farther away from the airstrip.

Explanation:

The problem involves two planes with congruent path lengths (150 miles and 220 miles, swapped between planes). For the first plane, the included angle between its two paths is $180^\circ - 30^\circ = 150^\circ$. For the second plane, the included angle is $180^\circ - x^\circ$. Since $x < 30$, $180 - x > 150$. By the Hinge Theorem, a larger included angle between two sides of a triangle (here, the paths from the airstrip) results in a longer third side (distance from the airstrip). Thus, the second plane is farther.

Answer:

The second plane; the legs are congruent. If $x < 30$, then the measure of the included angle, $(180 - x)^\circ$, is greater for the second plane; so, by the Hinge Theorem, the second plane is farther away from the airstrip.