Math Challenge

This math challenge was easy to approach, because I realized that the lines O1O, O1O2, and O2O1, form a triangle, I realized that if that triangle was a right triangle I could find the distance between O1, and O. To prove that triangle OO1O2 is a right triangle I used the radii of O1, I realized that the sides of triangle PO1Q, O1P, and O1Q, are radii, and therefore they are equal. Since they are equal, the triangle PO1Q, is an isosceles triangle. In an isosceles triangle the base angles are congruent , so that means that angle Q is equal to angle P, since angle Q is 45 degrees, angle P is also 45, and because a triangle has 180 degrees, the remaining angle(O1) must be 90 degrees. This proves that the triangle PO1Q, is a right triangle, and …show more content…
Since angle O1 QP is 45 degrees, that means that the angle directly below it is 45 degrees as well, and the angles to the right, and left are 135 degrees. Next I drew a line perpendicular to the line continuation of line P, and named it A since two perpendicular lines, make 90 degree angles, that means that I could subtract 90 from 135 (45), and that would be the angle measure of the angle between line A, and line Q. If that angle is 45, then the angle diagonal from it is also 45.Next I drew a line perpendicular to line A, that also goes through the center of circle O1(line B) , this forms a right triangle because two perpendicular lines make a right angle. From this, I could infer that the angle of the triangle formed, that is closest to the center of O1, is 45 degrees. After that realization, I drew a continuation of the line OO1, and a continuation of the line O1O2 , I realized that the angle between line B, and line OO1, is 45 degrees, and so the angle to the right of it is 135 degrees. Line O1O2 forms a 180 degree angle, and it includes a 45 degree angle, and a fraction of a 135 degree angle, I subtracted 45 from 135, to get the remaining portion of the angle, and that is 90 degrees, so that means that it is a right