Modern triangle geometry
Hey kiddo, have you ever heard of a triangle? It's a shape with three sides and three corners, just like the ones you see in a lot of buildings or on a pizza. When we talk about modern triangle geometry, we mean how mathematicians study and understand triangles using new ways and techniques.
So, imagine you pick any three points in space and connect them with lines to form a triangle. Modern triangle geometry can tell us a lot about that triangle, like how long those lines are, how big or small the angles between them are, and how those lines connect to each other.
One of the ways we can talk about triangles is by looking at their different types, like equilateral (where all three sides are the same length), isosceles (where two sides are the same length), and scalene (where no sides are the same length). We can also talk about the angles in the triangle, like acute (all angles less than 90 degrees), right (one angle is exactly 90 degrees), and obtuse (one angle is bigger than 90 degrees).
Mathematicians also study how lines intersect or touch the sides of the triangle. We call these lines "segments" or "rays." Some examples include medians (lines that connect a corner of the triangle to the midpoint of the opposite side), altitudes (lines that are perpendicular to one of the sides and pass through the opposite corner), and angle bisectors (lines that divide an angle into two equal parts).
Another cool thing modern triangle geometry can do is help us figure out where points are inside or outside of the triangle. For example, we can use the "incenter" (the point where all three angle bisectors meet) to find the circle that is inside the triangle and touches all three sides. We can also use the "orthocenter" (the point where all three altitudes meet) to find the circle that goes through all three corners of the triangle.
There are lots of other things we can learn about triangles using modern triangle geometry, but hopefully, this gives you a little taste of what it's all about!
So, imagine you pick any three points in space and connect them with lines to form a triangle. Modern triangle geometry can tell us a lot about that triangle, like how long those lines are, how big or small the angles between them are, and how those lines connect to each other.
One of the ways we can talk about triangles is by looking at their different types, like equilateral (where all three sides are the same length), isosceles (where two sides are the same length), and scalene (where no sides are the same length). We can also talk about the angles in the triangle, like acute (all angles less than 90 degrees), right (one angle is exactly 90 degrees), and obtuse (one angle is bigger than 90 degrees).
Mathematicians also study how lines intersect or touch the sides of the triangle. We call these lines "segments" or "rays." Some examples include medians (lines that connect a corner of the triangle to the midpoint of the opposite side), altitudes (lines that are perpendicular to one of the sides and pass through the opposite corner), and angle bisectors (lines that divide an angle into two equal parts).
Another cool thing modern triangle geometry can do is help us figure out where points are inside or outside of the triangle. For example, we can use the "incenter" (the point where all three angle bisectors meet) to find the circle that is inside the triangle and touches all three sides. We can also use the "orthocenter" (the point where all three altitudes meet) to find the circle that goes through all three corners of the triangle.
There are lots of other things we can learn about triangles using modern triangle geometry, but hopefully, this gives you a little taste of what it's all about!