8.4 The Distance Between Points A final application of the Pythagorean Theorem is on the coordinate plane. We can easily find the distance between two points vertically or horizontally on a coordinate plane just by counting, but finding the exact distance diagonally we have not been able to do until now. The Distance between Any Two Points On a coordinate plane, we can now find the distance between any two points by drawing in a right triangle and using the Pythagorean Theorem. Consider the following example: ЭЂ Э„ вЂ«Э’вЂ¬ Notice that if we want to find the distance between these two points, б€є2,2б€» and б€є5,6б€», we need to find the length of ЭЂ. Also note that Э„ is the horizontal distance between the points and вЂ« Э’вЂ¬is the vertical distance between the points. With all those values we now have a right triangle and can use the Pythagorean Theorem as follows: Э„а¬¶ аµ… вЂ« Э’вЂ¬а¬¶ аµЊ ЭЂ а¬¶ 3а¬¶ аµ… 4а¬¶ аµЊ ЭЂ а¬¶ 9 аµ… 16 аµЊ ЭЂа¬¶ 25 аµЊ ЭЂа¬¶ 5аµЊЭЂ So we know that the distance between these points is five units. While this is easy to see when drawn out on the coordinate plane, there are times when we are given the two points without a picture. In that case, we have two options. We can either draw the points on the coordinate plane as above, or we can find the horizontal and vertical distance between the points in another way. To do this without graphing, we realize that the horizontal distance between two points is the difference in their вЂ« Э”вЂ¬values. Why is this? Similarly, the vertical distance between two points is the difference in their вЂ«Э•вЂ¬ values. Again, can you explain why? So letвЂ™s look at our two points again, б€є2,2б€» and б€є5,6б€». The horizontal distance would be the difference between 2 and 5. Since difference means subtract, we can take 5 аµ† 2 аµЊ 3 to find the horizontal distance is 3. Similarly we can subtract the вЂ« Э•вЂ¬values to get 6 аµ† 2 аµЊ 4 meaning a vertical distance of 4. We can then plug in 3 and 4 into the Pythagorean Theorem and solve exactly as above. 278 Enrichment: The Distance Formula Using the information above, how would we find the distance between two generic points? We typically represent generic points with the notation of б€євЂ«Э”вЂ¬а¬µ , вЂ«Э•вЂ¬а¬µ ) and (вЂ«Э”вЂ¬а¬¶ , вЂ«Э•вЂ¬а¬¶ ). So what would the horizontal and vertical distance between these two points be? Horizontal distance: в„Ћ = вЂ«Э”вЂ¬а¬¶ в€’ вЂ«Э”вЂ¬а¬µ Vertical distance: вЂ«Э• = Э’вЂ¬а¬¶ в€’ вЂ«Э•вЂ¬а¬µ Finally, letвЂ™s substitute these into the Pythagorean Theorem of в„Ћа¬¶ + вЂ« Э’вЂ¬а¬¶ = ЭЂа¬¶ as follows and then solve for ЭЂ since ЭЂ is the actual distance between the points. (вЂ«Э”вЂ¬а¬¶ в€’ вЂ«Э”вЂ¬а¬µ )а¬¶ + (вЂ«Э•вЂ¬а¬¶ в€’ вЂ«Э•вЂ¬а¬µ )а¬¶ = ЭЂа¬¶ а¶Ґ(вЂ«Э”вЂ¬а¬¶ в€’ вЂ«Э”вЂ¬а¬µ )а¬¶ + (вЂ«Э•вЂ¬а¬¶ в€’ вЂ«Э•вЂ¬а¬µ )а¬¶ = а¶ҐЭЂа¬¶ а¶Ґ(вЂ«Э”вЂ¬а¬¶ в€’ вЂ«Э”вЂ¬а¬µ )а¬¶ + (вЂ«Э•вЂ¬а¬¶ в€’ вЂ«Э•вЂ¬а¬µ )а¬¶ = ЭЂ The final result is what is known as the distance formula. LetвЂ™s use this formula to find the distance between the points (в€’3, 4) and (3, в€’4). ЭЂ = а¶Ґ(вЂ«Э”вЂ¬а¬¶ в€’ вЂ«Э”вЂ¬а¬µ )а¬¶ + (вЂ«Э•вЂ¬а¬¶ в€’ вЂ«Э•вЂ¬а¬µ )а¬¶ ЭЂ = а¶Ґ(3 в€’ (в€’3))а¬¶ + ((в€’4) в€’ 4)а¬¶ ЭЂ = а¶Ґ(6)а¬¶ + (в€’8)а¬¶ ЭЂ = в€љ36 + 64 ЭЂ = в€љ100 ЭЂ = 10 We see that the distance between those two points is ten units. While the distance formula works, it is often easier to simply visualize the horizontal and vertical distance between two points mentally or on a coordinate plane. The distance formula is basically a fancy way to use the Pythagorean Formula and is meant for enrichment only. 279 Lesson 8.4 Determine the distance between the given points. Round your answers to three decimal places if necessary. 280 1. б€є1, 3) and (4, 7) 2. 2. (в€’3, 3) and (2, в€’9) 3. б€єаµ†2, в€’5) and (3, в€’8) 4. (в€’3, в€’3) and (3, 3) 5. б€є3, в€’2) and (5, 0) 6. (в€’3, в€’9) and (в€’3, 9) 7. б€є2, 1) and (3, в€’3) 8. (4, в€’2) and (7, 2) 9. б€є1, 1) and (7, 9) 10. (в€’8, 2) and (6, 2) 11. б€єаµ†4, 6) and (6, 2) 12. (2, 4) and (5, в€’2) 13. б€єаµ†5, в€’3) and (6, 6) 14. (в€’5, 4) and (7, 3) 281 282 15. б€єаµ†9, в€’3) and (в€’4, 4) 16. (2, в€’4) and (5, 4) 17. б€є0, 7) and (4, 2) 18. (в€’8, 7) and (7, в€’5)

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