1. Coordinate Proof Using Distance with Segments and Triangles p 521
CCSS GPE
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio

2. Warm up
1. What is the Pythagorean theorem?
2. Find x. x 3 4

3. Explore: Deriving the Distance Formula and the Midpoint Formula
A. To derive the Distance Formula, start with points J and K as shown in the figure.

4. Given: J(x1, y1) and K(x2, y2) with x1 ≠ x2 and y1 ≠ y2,
Prove: JK = √(x2 – x1)2 + (y2 – y1)2
What are the coordinates of point L? (JK is the hypotenuse of right triangle JKL).
Find JL
Find LK
By the Pythagorean Theorem, JK2 = JL2 + LK2

5. B To derive the Midpoint Formula
Given: A(x1, y1) and B(x2, y2)
Prove: The midpoint of AB is M

6. What is the horizontal distance from point A to point B?
What is the vertical distance from point A to point B?

7. The horizontal and vertical distances from A to M must be half these distances.
What is the horizontal distance from point A to point M?
What is the vertical distance from point A to point M?

8. To find the coordinates of point M, add the distances from Step E to the x- and y-coordinates of point A and simplify.

11. Reflections
Distance formula:
Midpoint formula: