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Question

a circle
Let A and B be the endpoints of a diameter of a circle. Let C be any point of the same circle. Show that the segments CA and CB are perpendicular.

Hint: I recommend using vector v =vector AM and vector w = vector MC

Put a point D on the circle so that AC is the same as BD.

Since M is in the midpoint of AB and the midpoint of CD,
so angle BMC is congruent to angle AMD and angle BMD is congruent to angle AMC.

This shows that CMA, BMC, BMD, and DMA are all isoceles triangles.
This says that the angle ACM is the same as angle CAM,
angle MCB is the same as angle BCM,
angle MAD is the same as angle MDA, and
angle MBD is the same as angle MDB.

Next, note that angle AMC + angle CMB = 180° since AB is a straight line.
Since angles CAM, ACM, and AMC make up a triangle, their sum is also 180°.
Since angles MCB, CBM, and CMB make up a triangle, their sum is also 180°.

Since they are isoceles triangles, angle CAM = angle ACM,
so we have 2(angle ACM) + angle CMA = 180°.

We also have angle CBM = angle BCM, so 2(angle BCM) + angle BMC = 180°.

Adding them together gives 2(angle ACM) + angle CMA + angle BMC + 2(angle BCM) = 360°.

It was said that angle(CMA) + angle(MBC) form the line defined by the segment AB,
so the sum of those two angles is 180°.  Subtracting this off gives
2(angle ACM) + 2(angle BCM) = 180°.

Dividing by 2 gives angle ACM + angle BCM = 90°.

This says that the angle ACB is 90°.

Volunteer

#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.