# Analytic Proofs of Theorems Previously Proved by Synthetic Means

### New York State Common Core Math Geometry, Module 4, Lesson 13

Student Outcomes

• Using coordinates, students prove that the intersection of the medians of a triangle meet at a point that is two-thirds of the way along each median from the intersected vertex.
• Using coordinates, students prove the diagonals of a parallelogram bisect one another and meet at the intersection of the segments joining the midpoints of opposite sides.

Analytic Proofs of Theorems Previously Proved by Synthetic Means

Classwork

Exercises

Let π΄(30,40), π΅(60,50), and πΆ(75,120) be vertices of a triangle.
a. Find the coordinates of the midpoint π of π΄π΅ and the point πΊ1 that is the point one-third of the way along ππΆ, closer to π than to πΆ.
b. Find the coordinates of the midpoint π of π΅πΆ and the point πΊ2 that is the point one-third of the way along ππ΄, closer to π than to π΄.
c. Find the coordinates of the midpoint π of πΆπ΄ and the point πΊ3 that is the point one-third of the way along ππ΅, closer to π than to π΅.

Exercise 1

a. Given triangle π΄π΅πΆ with vertices π΄(π1, π2), π΅(π1, π2), and πΆ(π1, π2), find the coordinates of the point of concurrency of the medians.
b. Let π΄(β23, 12), π΅(13, 36), and πΆ(23,β1) be vertices of a triangle. Where will the medians of this triangle intersect? (Use βTylerβs formulaβ from part (a) to complete this problem.

Exercise 2

Prove that the diagonals of a parallelogram bisect each other)

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