Similar Triangles in Circles and Right Triangles - Concept
Two triangles in a circle are similar if two pairs of angles have the same intercepted arc. Sharing an intercepted arc means the inscribed angles are congruent. Since these angles are congruent, the triangles are similar by the AA shortcut. If an altitude is drawn from the right angle in a right triangle, three similar triangles are formed, also because of the AA shortcut.
If you see a problem that looks like this, the question is do we have similar triangles. Well let's go back to what we know about inscribed angles in a circle.
If I pick one of these angles here, and I looked at the endpoints well that would be one endpoint right here, and one endpoint right there. The intercepted arc extends from one point to the other. Now there's another angle that has the exact same end points. So if these two have the exact same intercepted arc, then they must be congruent. The same could be said for these angles up here. They have the same intercepted arc. And last we have vertical angles, which means that these two must be congruent as well. So we have angle angle angle as congruent between these two triangles. Which means they must be similar. So if you see a problem like this and you're trying to find some of your side lengths, you know that you have similar triangles so you can set up proportions.
Let's look at one other special case. And that is if I have a right triangle and if from this right angle, if I dropped an altitude to the other side. What I'm going to do is I'm going to create a certain number of similar triangles. I'm going to redraw the two triangles that I've created down below. So I had created one triangle and the left side of that altitude and on the right side I've created another smaller triangle.
So if I look at this large triangle and I count that as triangle number 1, this is triangle number 2 and this is triangle number 3, I see that comparing triangle number 1 which is the large one, I have one right angle in each of these, and they share this angle right there which means you can use your angle angle shortcut to say that theses two triangles must be similar. The same thing can apply to this triangle on the right. Not only do both of these triangles have a right angle, but they share this angle in the corner. And I'm going to use two different markings.
So how many similar triangles have you created? We have three triangles that are all similar to each other. So remember these two key things, when you are looking at your test or your quiz.