what is the measure of the arc of the fountain that Charles lines of sight intersect?

Question

<!--[if !supportLists]-->1.    <!--[endif]-->Charles stands 18 m from a circular fountain. If his lines of sight form tangents to the fountain and make an angle of  45 degrees, what is the measure of the arc of the fountain that Charles lines of sight intersect? Draw/illustrate the problem

 

 

Answer 1

Let r be the radius of the fountain then r+18 metres is the distance between Charles (C) and the centre of the fountain, O. The lines of sight make tangents on either side of the fountain at points A and B. So triangles OAC and OBC are congruent right triangles, with right angles at A and B. Let r=radius of the fountain=OA=OB. ∠ACB=45° (given) so ∠ACO=∠BCO=½∠ABC=22.5°. sinACO=sinBCO=r/(r+18). Therefore, r=(r+18)sinACO.

r=(r+18)sinACO=rsinACO+18sinACO, r=18sinACO/(1-sinACO)=11.1585m approx. ∠AOB=2∠AOC=2(90-∠ACO)=135° or ¾π radians. Arc AB=¾πr=26.29m approx.

In the picture the blue circle is the fountain with centre O. Charles is at point C and the picture is to scale. AC and BC are lines of sight.