The lengths of the two heights of the unequal triangle ABC are 4 and 12 respectively. If the length of the third height is also an integer, then the length of this height is equal to______ ．

The lengths of the two heights of the unequal triangle ABC are 4 and 12 respectively. If the length of the third height is also an integer, then the length of this height is equal to______ ．

Because the lengths of the two heights of the unequal triangle ABC are 4 and 12 respectively, the lengths of the two sides of the triangle ABC can be set as 3x, X according to the equal area 3x × 4 = 12 × x (twice the area), s = 6x, because we know the assumed length of the two sides, according to the sum of the two sides is greater than the third side, and the difference between the two sides is less than the third side, we can get: 2x < the length of the third side < 4x, because it requires a higher maximum length, so when the third side is the shortest, the higher the height on the third side is, s = the length × height of the third side, 6x > 12 × 2x × height, 6 > height Equilateral triangles take the integer 5 as high, so the answer is: 5