c From Kahuku to Waimanolo: 20 miles From Nanakuli to Kahuku: 21 miles From Waimanolo to Nanakuli: 27 miles
a Let's put the information we are given on the diagram.
Source: OpenStreetMap
Notice that all angles of the triangle △ ABC are expressed in terms of m∠ B. Let's use the Triangle Angle-Sum Theorem to set up and solve an equation for m∠ B.
We can use this value and the expressions m∠ A=2+m∠ B and m∠ C=2(m∠ B)-14 to find the measure of the other two angles.
Angle
Expression
Measure
∠ A
2+48
m∠ A=50
∠ B
48
m∠ B=48
∠ C
2(48)-14
m∠ C=82
b Let's replace the expressions with the measure of the angles we got in part A. Let's also replace the label of the points with the name of the cities.
We can use the Angle-Side Relationships in Triangles to compare the lengths of the three legs of Kailey's trip.
The shortest leg is opposite to the angle with smallest measure, so it is between Kahuku and Waimanalo.
The next leg is the one opposite to Waimanalo, so it is between Nanakuli and Kahuku.
The longest leg is opposite to the angle with greatest measure, so it is between Waimanalo and Kahuku.
c To be able to use algebra to answer this part, let's introduce variables and write equations to express the information given in the question in terms of these variables.
Information
Variable/Expression
Length of the shortest leg.
s
Length of the middle leg.
m
Length of the longest leg.
l
The middle leg is 11 miles greater than one-half the length of the shortest leg.
m=1/2s+11
The longest leg is 12 miles greater than three-fourths of the shortest leg.
l=3/4s+12
The length of the entire trip is 68 miles.
s+m+l=68
Notice that we have an expression for both l and m in terms of s. We can use these expressions to replace l and m in the last equation to set up and solve an equation for s.
