Найдём координаты векторов \(\overrightarrow {AB} \) и \(\overrightarrow {CB} \):
\(\overrightarrow {AB} = \left( {13-5;y-1} \right) = \left( {8;y-1} \right);\,\,\,\,\,\,\,\,\overrightarrow {CB} = \left( {13-2;y-4} \right) = \left( {11;y-4} \right).\)
Длина вектора \(\vec d\left( {x;y} \right)\) равна: \(\left| {\vec d} \right| = \sqrt {{x^2} + {y^2}.} \)
Тогда: \(\left| {\overrightarrow {AB} } \right| = \sqrt {64 + {{\left( {y-1} \right)}^2}} ;\,\,\,\,\,\,\,\,\left| {\overrightarrow {CB} } \right| = \sqrt {121 + {{\left( {y-4} \right)}^2}} .\)
Так как по условию \(AB = CB\), то:
\(\sqrt {64 + {{\left( {y-1} \right)}^2}} = \sqrt {121 + {{\left( {y-4} \right)}^2}} \,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,64 + {y^2}-2y + 1 = 121 + {y^2}-8y + 16\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,\,\,\,6y = 72\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,y = 12.\)
Ответ: 12.