Unit 3 - 7
Key Objectives (from the notes):
- Prove and apply theorems about perpendicular bisectors;
- Prove and apply theorems about angle bisectors.
I don't know how, but I think the "key objectives" from the notes are just plain inaccurate. But anyway, welcome to the seventh lesson! In some places, 7 is referred to as a lucky number, so how lucky are you?
In this lesson, the main topic was very emphasized: triangle centers! We learned that there are hundreds and hundreds of different triangle centers, and we also covered the biggest four centers: incenter, circumcenter, centroid, and orthocenter.
Basically, each of these centers is the point of concurrency for three lines. The incenter pertains to the three angle bisectors, the circumcenter pertains to the three perpendicular bisectors, the centroid pertains to the three medians, and the orthocenter pertains to the three altitudes. You might be wondering, "The what now?" because some of you probably didn't recognize two words of that previous sentence. Well, that's why we have a glossary on this webpage, too, and for this unit, Oliver Shi is doing it, so you can be sure that there will be quality work going into those definitions!
That's basically the gist of it. Some other things that may bring up concern are the Centroid Theorem, the Perpendicular Bisector Theorem and its converse, and the Angle Bisector Theorem and its converse. Below is a picture and explanation for each:
In this lesson, the main topic was very emphasized: triangle centers! We learned that there are hundreds and hundreds of different triangle centers, and we also covered the biggest four centers: incenter, circumcenter, centroid, and orthocenter.
Basically, each of these centers is the point of concurrency for three lines. The incenter pertains to the three angle bisectors, the circumcenter pertains to the three perpendicular bisectors, the centroid pertains to the three medians, and the orthocenter pertains to the three altitudes. You might be wondering, "The what now?" because some of you probably didn't recognize two words of that previous sentence. Well, that's why we have a glossary on this webpage, too, and for this unit, Oliver Shi is doing it, so you can be sure that there will be quality work going into those definitions!
That's basically the gist of it. Some other things that may bring up concern are the Centroid Theorem, the Perpendicular Bisector Theorem and its converse, and the Angle Bisector Theorem and its converse. Below is a picture and explanation for each:
The Perpendicular Bisector Theorem states that a point (in the picture, this is X) that lies on a perpendicular bisector of a segment will always be the same distance from the endpoints of said segment, no matter how long the segment and no matter where the point is, as long as it is on the perpendicular bisector. Don't forget that the converse is also true here.
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The Angle Bisector Theorem states that a point (in the picture, this is C) that lies on an angle bisector of an angle will always be the same distance (assuming you're using the shortest one) from the sides of said angle, no matter how large the angle and no matter where the point is, as long as it is on the angle bisector. Don't forget that the converse is also true.
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That about covers it for this lesson.