Coordinate Proof Using Distance with Segments and Triangles p 521

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Amy Hatfield Central High School

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Presentation transcript:

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

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

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.

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

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

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

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?

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

Reflections Distance formula: Midpoint formula: