If three sides a, B, C of triangle ABC satisfy the condition: A ^ 2 + B ^ 2 + C ^ 2 + 338 = 10A + 24B + 26, then the height of the longest side of the triangle is___

If three sides a, B, C of triangle ABC satisfy the condition: A ^ 2 + B ^ 2 + C ^ 2 + 338 = 10A + 24B + 26, then the height of the longest side of the triangle is___

A ^ 2 + B ^ 2 + C ^ 2 + 338 = 10A + 24B + 26c a ^ 2-10a + 5 ^ 2 + B ^ 2-24b + 12 ^ 2 + C ^ 2-26c + 13 ^ 2 = 0 (a-5) ^ 2 + (B-12) ^ 2 + (C-13) ^ 2 = 0, so a = 5; b = 12; C = 13.5 ^ 2 + 12 ^ 2 = 13 ^ 2, so it is a right triangle, so triangle area = 1 / 2 × 5 × 12 = 1 / 2 × 13 × height, so it is high

If three sides a, B and C of △ ABC satisfy A2 + B2 + C2 + 338 = 10A + 24B + 26c, then the area of △ ABC is ()
A. 338B. 24C. 26D. 30

From A2 + B2 + C2 + 338 = 10A + 24B + 26c, it is obtained that: (a2-10a + 25) + (b2-24b + 144) + (c2-26c + 169) = 0, that is: (a-5) 2 + (B-12) 2 + (C-13) 2 = 0, a-5 = 0, B-12 = 0, C-13 = 0, a = 5, B = 12, C = 13, ∵ 52 + 122 = 169 = 132, that is, A2 + B2 = C2, ∠ C = 90 °, that is, the triangle ABC is a straight angle triangle. S △ ABC = 12 × 5 × 12 = 30