Question number 16. A rear windshield wiper has an arm R and blade P which meet at an angle alpha as illustrated. The wiper pivots about the point C and cleans the indicated region. What is the measure of the angle beta between the line tangent to the top arc and the rightmost vertical boundary of the indicated region? So let us begin by labeling the edges that have the same length, that would be P and R because we're looking at an isosceles triangle. And let's draw in the third side of that triangle like so. And let's call the two angles that are not labeled gamma. Since that's an isosceles triangle they're the same angle, both gamma. After rotation, where do these end up? So the blade P has to become the rightmost vertical boundary of the indicated region and it connects to and it connects to C like so by the arm R. So let's label these. This is R and this is P. Let's draw in also the missing third side and label the angles again. We have the angle alpha and then the angle gamma here and again over here. So we note that the orange third side of the isosceles triangle is exactly the radius of the circle centered at P and anything tangent to that circle, so I mean the green line here, will have to be perpendicular to the radius and we can find a 90 degree angle here in our picture and that's exactly this angle here like so. So beta is therefore pi over 2, the 90 degree angle, plus gamma. Gamma twice that plus alpha has to be equal to pi. So now we can solve for gamma and substitute. Gamma is equal to pi minus alpha divided by 2 and so beta is pi over 2 plus pi minus alpha divided by 2 which gives us pi minus alpha over 2 and so that is answer B.

geometry problem

angle beta

isosceles triangle

perpendicularity

solution