What do the crew members know about the triangles?
B
b
Look for corresponding sides in the triangles.
A
a
See solution.
B
b
No; see solution.
Practice makes perfect
a
We want to know the distance across the lake, FG, so let's look at the triangles the crew team formed to determine it. The positioning of the vertices of the triangles guarantee that â–³ FGK and â–³ JHK are congruent triangles. See the proof of this claim in part B.
Using congruence of sides, the crew can find the distance across the lake by measuring the corresponding distance in triangle â–³ JHK. See the details in part B.
b
On the diagram below we used the same color for two vertices if they are collinear with vertex K.
Let's summarize the relationship we can determine from the diagram about triangles â–³ F G K and â–³ J H K.
Looking for Congruent Sides
The markers on the diagram indicate a congruent side pair.
K F≅ K J
Looking for Congruent Angles
The markers on the diagram indicate two right angles. Since all right angles are congruent, this indicates a congruent angle pair.
∠K F G≅ ∠K J H
We can also see that the angles at K are nonadjacent angles formed by two intersecting lines. These are vertical angles, and hence congruent.
∠G K F≅ ∠H K J
Concluding Congruence
We now know that in triangles â–³ F G K and â–³ J H K two angles and the included side are congruent.
According to the Angle-Side-Angle (ASA) Congruence Postulate, this means that the two triangles are congruent.
△ F G K≅△ J H K
Conclusion
The crew is interested in the length of F G.
The side of â–³ J H K corresponding to F G is J H.
F G≅ J H
We know that corresponding sides of congruent triangles are congruent, and congruent segments have the same measure. This means that the given measurement J H=1350m also gives the length of F G.
F G=1350
Since 1350<1500, the crew can conclude that the lake is not long enough to use as a location for their regatta.
