Question

Consider this scenario and then answer the next three questions: A land owner wants a good approximation of the width of his stream from point D to point C. At high point B, he lays a carpenter’s square (used to make right angles) so that he can sight along BC and BA. AD = 9.3 ft and BD = 15.7 ft.1) State the corresponding proportional sides between similar triangles ABC and ADB. 2) Calculate the length of segment AB. Round the answer to the nearest tenth. (Hint: Use Pythagorean Theorem)3) Answer the following questions about the previous scenario: (a) Write an algebraic expression to represent the length from A to C. (b) Write a proportion to find the distance across the stream. (c) What is the length across the stream?

Answer 1

1) Let's consider both triangles ABC and ADB:

Then the proportional sides between similar triangles is:

`(AB)/(AD)=(BC)/(BD)=(AC)/(AB)`

2)Since we have that AD=9.3 and BD=15.7, using the pythagorean theorem we get the following:

`\begin{gathered} AB^2=AD^2+DB^2 \\ \Rightarrow AB^2=(9.3)^2+(15.7)^2=86.5+246.5=333 \\ \Rightarrow AB=\sqrt[]{333}=18.2 \\ AB=18.2 \end{gathered}`

Therefore, AB=18.2 ft

3)a)the best way to represent the length from A to C is with the pythagorean theorem:

`AC=\sqrt[]{AB^2+BC^2}`

b)The proportion to find the distance across the stream is the segment DC, then:

`(AD)/(BD)=(DB)/(DC)`

c)We can find the length by using the values for AD, BD and DB that we previously got:

`\begin{gathered} AD=9.3 \\ BD=15.7 \\ \Rightarrow(9.3)/(15.7)=(15.7)/(DC) \\ \Rightarrow DC\cdot((9.3)/(15.7))=15.7 \\ \Rightarrow DC=(15.7)/(((9.3)/(15.7)))=26.5 \\ DC=26.5 \end{gathered}`

Finally, we have that the length across the stream is 26.5 ft