# Analytic Proofs of Theorems Previously Proved by Synthetic Means

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

Worksheets for 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 𝐴(30,40), 𝐵(60,50), and 𝐶(75,120) be vertices of a triangle.
a. Find the coordinates of the midpoint 𝑀 of 𝐴𝐵 and the point 𝐺1 that is the point one-third of the way along 𝑀𝐶, closer to 𝑀 than to 𝐶.
b. Find the coordinates of the midpoint 𝑁 of 𝐵𝐶 and the point 𝐺2 that is the point one-third of the way along 𝑁𝐴, closer to 𝑁 than to 𝐴.
c. Find the coordinates of the midpoint 𝑅 of 𝐶𝐴 and the point 𝐺3 that is the point one-third of the way along 𝑅𝐵, closer to 𝑅 than to 𝐵.

Exercise 1

a. Given triangle 𝐴𝐵𝐶 with vertices 𝐴(𝑎1, 𝑎2), 𝐵(𝑏1, 𝑏2), and 𝐶(𝑐1, 𝑐2), find the coordinates of the point of concurrency of the medians.
b. Let 𝐴(−23, 12), 𝐵(13, 36), and 𝐶(23,−1) be vertices of a triangle. Where will the medians of this triangle intersect? (Use “Tyler’s formula” from part (a) to complete this problem.

Exercise 2

Prove that the diagonals of a parallelogram bisect each other)

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