Pedro starts from the camp C, walks east 5 miles to P, turns 15^(∘) south of east, and walks 2 more miles to the waterfall W.
Joel starts from the camp C, walks west 5 miles to J, turns 35^(∘) north of west, and walks 2 more miles to the lake L.
To compare the distances of the destinations from the camp, CW and CL, let's consider triangles △ CPW and △ CJL.
The two triangles have two pairs of congruent sides.
Congruent Sides
Justification
CP≅CJ
Both segments represent a 5 kilometer walk.
PW≅JL
Both segments represent a 2 kilometer walk.
Let's compare the angles at P and at J.
We can find the measure of these angles using that they form a linear pair with the respective turning angles.
15+ m∠ P=180 &⟹ m∠ P=165
35+ m∠ J=180 &⟹ m∠ J=145
We now know that triangles △ CPW and △ CJL have two pairs of congruent sides, and the included angle in triangle △ CPW is larger than the included angle in triangle △ CJL.
m∠ P> m∠ J
According to the Hinge Theorem, this implies that the third side of triangle △ CPW is longer than the third side of triangle △ CJL.
CP> CL
When Pedro and Joel reach their destinations, Joel is closer to the camp.
b Let's modify the diagram from Part A and use a similar method.
Now we will compare the angles at P and J.
15+ m∠ P=180 &⟹ m∠ P=165
10+ m∠ J=180 &⟹ m∠ J=170
As in Part A, triangles △ CPW and △ CJL have two pairs of congruent sides.
This time the included angle in triangle △ CPW is smaller than the included angle in triangle △ CJL.
m∠ P< m∠ J
According to the Hinge Theorem, this implies that the third side of triangle △ CPW is shorter than the third side of triangle △ CJL.
CP< CL
If Joel turned 10^(∘) south of west instead of 35^(∘) north of west, Joel would be farther from the camp.
