Part a: The triangles ∆ABC and ∆EDC are similar by AA similarity rule.
Part b: The width of AB is 67.5 feet
Explanation:
Part a: We need to prove that the two triangles ABC and EDC are similar.
To prove the triangles are similar, then their angles must be similar.
Thus, we have,
∠DCE and ∠BCA are similar (vertical angles)
∠CDE and ∠CBA are similar (right angles)
∠B and ∠A are similar
Hence, the triangles ∆ABC and ∆EDC are similar by AA similarity rule.
Part b: We need to determine the width of AB
Since, the triangles are similar, then their corresponding lengths are proportional.
Thus, we have,
![\frac{DE}{AB} =\frac{DC}{CB}](https://tex.z-dn.net/?f=%5Cfrac%7BDE%7D%7BAB%7D%20%3D%5Cfrac%7BDC%7D%7BCB%7D)
where
,
and ![CB=90ft](https://tex.z-dn.net/?f=CB%3D90ft)
Substituting these values, we get,
![\frac{54}{AB} =\frac{72}{90}](https://tex.z-dn.net/?f=%5Cfrac%7B54%7D%7BAB%7D%20%3D%5Cfrac%7B72%7D%7B90%7D)
Multiplying both sides by 90, we get,
![\frac{4860}{AB}=72](https://tex.z-dn.net/?f=%5Cfrac%7B4860%7D%7BAB%7D%3D72)
![\frac{4860}{72}=AB](https://tex.z-dn.net/?f=%5Cfrac%7B4860%7D%7B72%7D%3DAB)
Dividing, we have,
![67.5=AB](https://tex.z-dn.net/?f=67.5%3DAB)
Thus, the width of the river AB = 67.5 feet