Distance on a Coordinate Grid 4.11

How far? Jenn’s house is 4 miles west and 4 miles north of the center point of the city. Viet’s house is 1 mile south of the center point of the city. How many miles is it to go straight from Jenn’s house to Viet’s house?

Coordinate Grid Today, we are attempting to determine the distance between two points on a coordinate grid Sounds unrelated to what we’ve been doing, huh? Give me a minute and you’ll see…

Distance Formula This is the official formula: 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 Doesn’t that look fun to memorize and use? No? Would you like an easier way?

The easier way… It starts with finding how much you go up/down and over to get from one point to the next Sound familiar? That’s right…it’s just slope!

Use what you know about slope… The first step is to find the difference in the x-values and the difference in the y-values Just like you do in slope! (-4, 4) and (0, -1) 𝑚= −1 − (4) 0 − (−4) = −5 4 Then use this to label the legs of the right triangle… -5 4

Now, to find that missing side…hmmm… How do you find the missing side of a right triangle? Pythagorean Theorem!! I told you it was all related!

Let’s finish this off! The legs of the triangle are -5 and 4 (Don’t worry about that negative…it just tells us that the next point went down from the first…and when we square it, it goes away) (−5) 2 + (4) 2 = 𝑑 2 25 + 16= 𝑑 2 41= 𝑑 2 41 = 𝑑 2 𝑑≈6.4 Therefor, the distance between the two points is approximately 6.4 units!

How did they get that crazy formula? 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 = 𝑏 𝑎 ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 = 𝑑 2 ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 =𝑑 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2

Notes – Distance on a Coordinate Grid Distance between points is usually a diagonal line which means you cannot just count the units Step 1) Create a slope triangle between the points. (Do NOT simplify the slope!) Step 2) Use Pythagorean Theorem to determine the length of the hypotenuse of the slope triangle OR substitute the coordinates into the distance formula 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2

Distance on a Coordinate Grid Practice Calculate the distance between the points below. Round to the nearest tenth if necessary. 1) (-1, 3) and (2, 7) 2) (0, 2) and (8, -4) 3) (-7, -2) and (-3, 3) 4) (12, 20) and (3, 15) 5) To get to the playground from the entrance of the park, Laura had to walk 12 meters east and then 9 meters south. What is her exact distance from the entrance? 5 units 6.4 units 15 meters 10 units 10.3 units