Question 6

6. Write a proof of the result that an angle inscribed in a semi-circle is a right angle. You may use the result that the sum of the angles of a triangle is 180º.

To solve this problem, I have to first draw a circle, and then a triangle on half of it. The picture should look like the picture below:

I then named each corner of the triangle to be A,B, and C. Then, I put a center D on the circle and connected in to point B. Because, D is at the center of the circle - it means that any point on the circle from D to that point is the radius and they are all equal. This means:

AD = CD = BD

Then, let's look at the two smaller triangles created by BD.

Triangle ABD is an isoceles triangle - it has two sides that are equal. How do I know this? Two of it sides are AD, and BD, which I have stated earlier to be the same because they are both the radius of the circle. That means that the angles opposite of these two lines are also the same meaning that the orange angle (let's call it x) and red angle (let's call it y) is equal. x = y

I know that that the sum of all the angles in a triangle is = 180, so:

x + y + z = 180

z = 180 - x - y

Then, I repeated the same steps for the other triangle. Triangle CBD is also a isoceles triangle. I know this because has sides BD and CD which I have also stated earlier are equal to each other. This then means that the angles opposite of BD, and CD - purple angle l and brown angle n are equal to each other. l = n.

Then, I know that the sum of all the angles in a triangle is = 180, so:

l + m + n = 180

m = 180 - l - n

Looking at the picture, I know that triangle ABC is produced from "adding" these two smaller triangles together, so I can also add these two equations together:

(z = 180 - x - y ) + (m = 180 - l - n)

z + m = 360 - x - y - l - n

Then, what do I do next? I have to 6 variables and only three equations. Well, I know that l = n and x = y so let's get rid of two variables by replacing them with their equivalent. So, for my purposes, I will get rid of n and replace it with l, and get rid of x, and replace it with y:

z + m = 360 - y - y - l - l

z + m = 360 -2y - 2l

Then, I know angles z + m  180. Why? Because the two of them combined makes a straight line, and I know that straight lines are 180 degrees. So, let's replace z + m with 180 in the equation:

180 = 360 - 2y - 2l

2y + 2l + 180 = 360

2y + 2l = 180

y + l = 90

Angle y + angle l = angle b in the larger triangle, triangle ABC (I obtain this from looking at the picture). 

If angle y + angle l = 90, and angle y + angle l = angle b then:

angle b = 90.

If angle b is 90 degrees then an angle inscribed in a semi circle is a right angle.