The method of placing the stakes guarantees that triangles â–³ ABE and â–³ CDE are congruent. See the proof of this claim in part B.
By using congruence of sides, the surveyor can find the distances across the river (AB and BE) by measuring the corresponding distances in triangle â–³ CDE.
See the details in part B.
b
Let's follow the steps of placing the stakes during the surveying process and let's see what the method guarantees about the relationship of the two triangles. Vertices of the same color on the diagram below indicate correspondence, which we learn while making the diagram.
Placement of E
Since E is the midpoint between A and C it cuts segment AC into two congruent segments.
E A≅ E C
Placement of D
The final stake, D, is placed so that two conditions are satisfied.
C was placed so that CA⊥A B, and now D is placed so that C D⊥CA. We know that the angles at A and C are both right angles (and hence congruent).
∠E A B≅ ∠E C D
In placing D the surveyor also makes sure that D, E, and B are collinear. Since A, E, and C are also collinear, we know that the two nonadjacent angles formed at E are vertical, and hence congruent.
∠A E B≅ ∠C E D
Concluding Congruence
We now know that in triangles â–³ A B E and â–³ C D E two of their angles and the included side are congruent.
According to the Angle-Side-Angle (ASA) Congruence Postulate, this means that the two triangles are congruent.
△ A B E≅ ∠C D E
Finding AB
We know that corresponding sides of congruent triangles are congruent.
A B≅ C D
Congruent segments have the same measure.
This means that the measurement C D=550m also gives us the length of A B.
A B=550 meters
