pizer@ecsvax.UUCP (07/15/84)
References: <ecsvax.2861>,<oddjob.326> The following articles are both a summation and explanation of the solutions to the first three problems. I will explain the method I used to solve the problems, and then, if applicable, show the way other people solved the problem. I decided to split them up after the entire article reached a length of 100 lines. The answer to problem IV and problem V, and whatever follows will come later. #1 >>It is a semicircle with diameter CAF, with C & F being on the semicircle >>and A being the center. Point B is a point on the semicircle such that >>angle BAC is 40 degrees. Point E is a point on segment AF such that angle >>ABE is 10 degrees. Finally, point D is a point on the semicircle, and not >>point B such that angle ADE is also 10 degrees. Find angle DAF. My solution to this problem is to rotate triangle AED around the center, point A, until it overlaps triangle AEB. Then, lable point E on triangle AED point E', so there will be no confusion. It becomes apparent that since AE' and AE are congruent, triangle AEE' is isosceles, with angle E and E' being equal. Since you know angle BAE is 140 degrees (180-40), and that angle ABE is 10 degrees, angle AEB must be 30, along with angle AE'E. That leaves angle E'AE to be 180-30*2, or 120, thus making angle BAE' (also angle DAE', since D overlaps B) 20 degrees. Angle DAE', or just plain DAE when it is rotated back, is what we're looking for. Matt Crawford also posted a solution <oddjob.326 (if it is still around)>. He solved it using the law of sines to prove sin(AEB)=sin(AED), and since we are told point B is distinct from point D, angle AED must be 180 - AEB, or 150, making angle DAF 180-150-10, or 20 degrees. Another solution to this problem, and unfortunately I erased the letter accidently and can't remember his name, involves extending the semicircle to a full circle, and then extending BE till it hits the circle, labeled point G. Then, it becomes apparent that angle GAF is the reflection of angle DAF, and can then be solved with little trouble. I apologize to the author of this letter for not giving credit to him. Billy Pizer ({decvax,akgua,ihnp4,burl}!mcnc!ecsvax!pizer) -- "Why do the days get longer in the summer?" "Because heat makes things expand." --