The two sides of the triangle are 34 and 32, and the median to the third is 17. Find the area of the triangle.

1. The length of the median drawn to side a is found by the formula:
m = (√ (2b ^ 2 + 2c ^ 2 – a ^ 2)) / 2.
By hypothesis, b = 34, c = 32, m = 17.
Substitute the known values ​​into the formula and find the length of the third side of the triangle:
(√ (2 * 34 ^ 2 + 2 * 32 ^ 2 – a ^ 2)) / 2 = 17;
√ (2 * 1156 + 2 * 1024 – a ^ 2) = 34 (proportional);
2312 + 2048 – a ^ 2 = 1156;
4360 – a ^ 2 = 1156;
– a ^ 2 = 1156 – 4360;
– a ^ 2 = – 3204;
a ^ 2 = 3204;
a = √3204;
a = 6√89.
2. Area of ​​a triangle according to Heron’s formula:
S = √p (p – a) (p – b) (p – c),
where p is a semi-perimeter.
Semi-perimeter:
p = (a + b + c) / 2 = (6√89 + 34 + 32) / 2 = (6√89 + 66) / 2 = 3√89 + 33.
Find the area:
S = √ (3√89 + 33) (3√89 + 33 – 6√89) (3√89 + 33 – 34) (3√89 + 33 – 32) = √ (3√89 + 33) (33 – 3√89) (3√89 -1) (3√89 + 1) = √ (33 ^ 2 –
(3√89) ^ 2) ((3√89) ^ 2 – 1 ^ 2) = √ (1089 – 9 * 89) (9 * 89 – 1) = √ (1089 – 801) (801 – 1) = √288 * 800 = √230400 = 480.
Answer: S = 480.