Wzór na długość odcinka w układzie współrzędnych
[tex] |AB| = \sqrt{(x_2 - x_1 {)}^{2} + (y_2 - y_1 {)}^{2} } [/tex]
gdzie:
[tex]A = (x_1,y_1) \\ \\ B = (x_2,y_2)[/tex]
a)
[tex] |AB| = \sqrt{( - 2 - 2 {)}^{2} + (5 - 8 {)}^{2} } \\ |AB| = \sqrt{( - 4 {)}^{2} + ( - 3{)}^{2} } \\ |AB| = \sqrt{16 + 9} \\ |AB| = \sqrt{25} \\ |AB| = 5[/tex]
[tex] |BC| = \sqrt{(6 - ( - 2) {)}^{2} + ( - 1 - 5 {)}^{2} } \\ |BC| = \sqrt{(6 + 2{)}^{2} + ( - 6 {)}^{2} } \\ |BC| = \sqrt{ {8}^{2} + 36} \\ |BC| = \sqrt{64 + 36} \\ |BC| = \sqrt{100} \\ |BC| = 10[/tex]
[tex] |AC| = \sqrt{(6 - 2 {)}^{2} + ( - 1 - 8 {)}^{2} } \\ |AC| = \sqrt{ {4}^{2} + ( - 9 {)}^{2} } \\ |AC| = \sqrt{16 + 81} \\ |AC| = \sqrt{97} [/tex]
b)
[tex] |AB| = \sqrt{(5 - ( - 3) {)}^{2} + ( - 1 - 4 {)}^{2} } \\ |AB| = \sqrt{ {(5 + 3)}^{2} + ( - 5 {)}^{2} } \\ |AB| = \sqrt{ {8}^{2} + 25} \\ |AB| = \sqrt{64 + 25} \\ |AB| = \sqrt{89} [/tex]
[tex] |BC| = \sqrt{(5 - 5 {)}^{2} + (9 - ( - 1) {)}^{2} } \\ |BC| = \sqrt{ {0}^{2} + (9 + 1 {)}^{2} } \\ |BC| = \sqrt{0 + {10}^{2} } \\ |BC| = \sqrt{100} \\ |BC| = 10[/tex]
[tex] |AC| = \sqrt{(5 - ( - 3) {)}^{2} + (9 - 4 {)}^{2} } \\ |AC| = \sqrt{(5 + 3 {)}^{2} + {5}^{2} } \\ |AC| = \sqrt{ {8}^{2} + 25} \\ |AC| = \sqrt{64 + 25} \\ |AC| = \sqrt{89} [/tex]
c)
[tex] |AB| = \sqrt{(0 - ( - 2) {)}^{2} + (6 - 1 {)}^{2} } \\ |AB| = \sqrt{(0 + 2 {)}^{2} + {5}^{2} } \\ |AB| = \sqrt{ {2}^{2} + 25} \\ |AB| = \sqrt{4 + 25} \\ |AB| = \sqrt{29} [/tex]
[tex] |BC| = \sqrt{(6 - 0 {)}^{2} + (2 - 6 {)}^{2} } \\ |BC| = \sqrt{ {6}^{2} + ( - 4 {)}^{2} } \\ |BC| = \sqrt{36 + 16} \\ |BC| = \sqrt{52} \\ |BC| = 2 \sqrt{13} [/tex]
[tex] |AC| = \sqrt{(6 - ( - 2) {)}^{2} + (2 - 1 {)}^{2} } \\ |AC| = \sqrt{(6 + 4 {)}^{2} + {1}^{2} } \\ |AC| = \sqrt{ {10}^{2} + 1} \\ |AC| = \sqrt{100 + 1} \\ |AC| = \sqrt{101} [/tex]
