Use a two column proof to show that the centroid of an equilateral triangle divides the inside area into 6 congruent triangles
Since this is not my area of expertise, I searched Google under the key words "proof centroid of an equilateral triangle" to get this:
http://www.google.com/search?client=safari&rls=en&q=proof+centroid+of+an+equilateral+triangle&ie=UTF-8&oe=UTF-8
In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search. Also see http://hanlib.sou.edu/searchtools/.
that didn't really help...i was asking for a solution not a guide on how to search google.
To prove that the centroid of an equilateral triangle divides the inside area into six congruent triangles, we can use a two-column proof. Here's how we can do it:
Proof:
Statement Reason
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1. Let ABC be an equilateral triangle. Given
2. Let D, E, and F be the midpoints of sides AB, BC, and AC, respectively. Definition of midpoint
3. AD, BE, and CF are the medians of triangle ABC. Definition of median
4. The medians of a triangle intersect at a point called the centroid. Property of medians
5. Let G be the centroid of triangle ABC. Definition of centroid
6. DG, EG, and FG are the medians of triangle ABC. Definition of median
7. DG, EG, and FG intersect at point G. Property of medians
8. Triangle AGB is congruent to triangle BGC. SAS Postulate (Side-Angle-Side)
9. Triangle AGC is congruent to triangle CGA. SAS Postulate (Side-Angle-Side)
10. Triangle BGA is congruent to triangle CGA. SAS Postulate (Side-Angle-Side)
11. Triangle AGC is congruent to triangle AGC. Reflexive Property of Congruence
12. Triangle BGA is congruent to triangle BGA. Reflexive Property of Congruence
13. Triangle CGA is congruent to triangle CGA. Reflexive Property of Congruence
14. Triangle AGB is congruent to triangle BGA. Symmetric Property of Congruence
15. Triangle CGA is congruent to triangle BGA. Symmetric Property of Congruence
16. Triangle AGC is congruent to triangle BGC. Symmetric Property of Congruence
17. Triangle AGB is congruent to triangle CGA. Transitive Property of Congruence
18. Triangle BGA is congruent to triangle AGC. Transitive Property of Congruence
19. Triangle CGA is congruent to triangle AGB. Transitive Property of Congruence
20. All six triangles have congruent areas. Congruence of triangles implies congruent areas.
In this proof, we start by defining an equilateral triangle ABC. Then, we define the midpoints of its sides as D, E, and F. We establish that medians AD, BE, and CF intersect at a point called G, which is the centroid. Using congruence postulates, we prove that each of the six resulting triangles - AGB, BGA, CGA, BGC, AGC, and BCG - are congruent to each other. Therefore, they have congruent areas.