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# 12.2: Distances in the Coordinate Plane

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 11, Lesson 5.

In this activity, you will explore:

• finding the length of a segment using the Distance Formula
• finding the length of a segment using the Pythagorean Theorem.

## Problem 1 – The Distance Formula

Construct a segment. Find the coordinates of the endpoints and the measured length. Use the distance formula to calculate the length.

Endpoints Measured Length Calculated Length
(_____ , _____) and (_____ , _____) ______________ ______________
(_____ , _____) and (_____ , _____) ______________ ______________

What is important to remember when using the Distance Formula?

What happens to the Distance Formula when your segment is horizontal or vertical? Give an example using endpoints.

(_____ , _____) and (_____ , _____)

Problem 2 – The Distance Formula and the Pythagorean Theorem

Find the length of all three sides of your triangle. Which side is the longest? Can two of the sides be equal lengths? Which two?

Use the Pythagorean Theorem to calculate the length of your segment in another way.

Endpoints Measured Length pythagorean Length
(_____ , _____) and (_____ , _____) ______________ ______________
(_____ , _____) and (_____ , _____) ______________ ______________

What is the relationship between the Pythagorean Theorem and the Distance Formula?

## Problem 2 - Apply The Math

What formula gives the distance between the points (x1,y1)\begin{align*}(x_1, y_1)\end{align*} and (x2,y2)\begin{align*}(x_2, y_2)\end{align*}?

Determine the length of the segment with the following endpoints:

1. (1,2)\begin{align*}(1, 2)\end{align*} and (5,10)\begin{align*}(5, 10)\end{align*}

2. (5,8)\begin{align*}(5, 8)\end{align*} and (9,5)\begin{align*}(9, 5)\end{align*}

3. (7,4)\begin{align*}(7, 4)\end{align*} and (4,7)\begin{align*}(4, 7)\end{align*}

4. (2,3)\begin{align*}(-2, 3)\end{align*} and (3,5)\begin{align*}(3, 5)\end{align*}

5. (1,9)\begin{align*}(1, -9)\end{align*} and (2,7)\begin{align*}(-2, -7)\end{align*}

6. (3,5)\begin{align*}(3, 5)\end{align*} and (3,11)\begin{align*}(3, -11)\end{align*}

Given an endpoint and a length of a segment, find a possible other endpoint:

7. Endpoint: (3,1)\begin{align*}(3, 1)\end{align*}; Length 5\begin{align*}5\end{align*}

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