My frst essay will be working with trisections of the area of a triangle.
Triangle #1
Given a triangle ABC, find a point D such that line segments AD, BD, and CD trisect the area of the triangle into three regions with equal areas.
I will of course use GSP to work with this triangle.
So first i will draw a triangle and call it ABC.
Now the problem wants us to find a point D inside this triangle so that the segemnts formes with ABC form a trisection of the area of the triangle.
I first just put a point D in the triangle and made segments to ABC and just played around, moving d around untill the trisections were all equal. but there is a much easier and much faster way to do it.
First you need to find the midpoints of all three sides of your triangle
Then connect them with the points oppisite of them and make a point on the intersection of the three resulting lines and that is your D.
Then you will need to hide your segments and midpoints to get this
Color it in, measure your areas to see if you have three equal areas
And we see how to find a point D inside a triangle that when connected With ABC their segments form three triangles with the same area. But what exactly is point D and how do we prove that D and these segments form three equal triangles every time.
What is D? Point D is the intersection of the three segments formed by connecting the end points of a triangle with the midpoints of the opposite side.
What's the proof of this?
When you connect one of the arcs of the triangle with the midpoint of the opposite side it cuts the triangle into two parts of equal area
If you go on and connect all three arcs with the opposite midpoint the triangle will be broken in to six triangles with equal area
We know that each set of two trinagles sharing the same base of our original triangle have the same area because area of a triangle is 1/2 base time height. They have the same base(1/2 the base of our original triangle and the same hieght. so since all six triangles have equal areas then we know that when you put the two together that share a base you will have three triangles with the same area.