If in acute angled triangle ABC, AL, BM, and CN are the three altitudes of triangle ABC, then which of the following statements will be true?
Show answer & explanation
Correct answer: D
Explanation
The correct answer is option 4 i.e. AL + BM + CN < AB + BC + CA.
Evaluate each of the given options:
1) AL + BM + CN = AB + BC + CA
This statement suggests that the sum of the altitudes equals the perimeter of the triangle (sum of the sides). This is not true because the altitudes are generally shorter than the sides. Hence, the sum of the altitudes will be smaller than the sum of the sides.
2) AL + BM = AB + BC
This statement suggests that the sum of two altitudes equals the sum of two sides. This is also not true, as altitudes are typically shorter than their corresponding sides.
3) AL + BM + CN > AB + BC + CA
This statement suggests that the sum of the altitudes is greater than the perimeter of the triangle. This is not true, as the altitudes are generally shorter than their corresponding sides, so their sum will not exceed the perimeter.
4) AL + BM + CN < AB + BC + CA
This statement suggests that the sum of the altitudes is less than the perimeter of the triangle. This is true, as the altitudes are shorter than the sides of the triangle. Therefore, the sum of the altitudes will always be less than the perimeter.