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# Geometry SPECIAL PROPERTIES OF TRIANGLES

This is our last dance with triangles before we move on to more complicated shapes, so let’s recap.  We started out with learning the anatomy of a single triangle, looked at different types of triangles, and then moved on to comparing parts of triangles to determine congruency.  In the next unit we looked at right triangles and got a strong grasp on the Pythagorean theorem and trigonometric ratios.  Now we’re going to look at the pieces you get when you cut up triangles, and the orientation of triangles to other shapes by looking at specific concepts such as the midsegment of a triangle, the equation for the perpendicular bisector, and the enigmatic circumcenter.   If you’re solid with the earlier material, just download the study guide and print it out to refer to when you study with your friends.  If you’re not, you might need a geometry tutor and now is a good time to get in touch to make sure things don’t snowball on you.  Either way, it's important to see the illustrations  in the tutorial as the concepts in this unit are tough to visualize without it.

Here are few explanations to get you started.

## The Midsegment Theorem

The midsegment is essentially a line segment connecting the midpoints of two sides of a triangle. To grasp this concept fully, you need to rewind a little and remember that the midpoint of a line segment is the point that divides a segment into two equal parts. In the context of a triangle, the midsegment connects the midpoints of any two sides, creating a line that is parallel to the third side. By identifying and understanding midsegments, mathematicians and students gain a powerful tool for dissecting and analyzing triangles. It allows for the establishment of relationships between different parts of a triangle, paving the way for more advanced problem-solving techniques.

## The Perpendicular Bisector

The perpendicular bisector of a segment is a line that not only bisects the segment but does so at a right angle. This means that the line that is being bisected ,and the line that is bisecting it make a 90 degree angle where they meet.  The slopes of these two lines are negative reciprocals of each other and the equations governing them can be used to determine its intersection with other geometric elements. This knowledge proves invaluable in solving geometric problems and gaining a deeper comprehension of the relationships between different components of a triangle.  In the case of a triangle, the perpendicular bisector also sometimes cuts through the angle opposite the side it is bisecting.   In the case of an isosceles the angle is also bisected, but in many cases the perpendicular bisector may not touch the opposite angle at all.

## Circumcenter

The circumcenter  is the point where the perpendicular bisectors of the sides of a triangle meet.  Let's unpack this because there's a lot going on.  First, understand that the lengths of the individual bisectors are not equal.  Second, they all intersect at a point that is equidistant from each of the vertices of the triangle.  Depending on the type of triangle, this point could be inside the triangle, outside the triangle, or on one of the sides of the triangle. If the triangle is circumscribed  by (surrounded by) a circle, the circumcenter is the centerpoint on that circle providing a geometric anchor for the triangle.

Having a solid grasp on understanding these ideas and the skills needed to work with them are particularly useful in real-world scenarios where geometric principles play a crucial role, such as in architecture, engineering, and physics. Consider a scenario where an architect needs to design a structurally sound building. Understanding the midsegments of various components can help in distributing loads evenly, ensuring stability. The equation for the perpendicular bisector becomes instrumental in determining optimal weight distribution, contributing to the overall structural integrity of the building. The concept of the circumcenter aids in creating well-balanced designs, aligning with the principles of symmetry and equilibrium.

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