Understanding the Problem: Altitudes in an Acute Triangle

The question asks us to compare the sum of the lengths of the altitudes of an acute-angled triangle ABC with the sum of the lengths of its sides. We are given that AL, BM, and CN are the altitudes from vertices A, B, and C respectively.

Let the lengths of the sides opposite to vertices A, B, and C be denoted by $a, b, c$ respectively. So, BC = $a$, AC = $b$, and AB = $c$. The lengths of the altitudes AL, BM, and CN are commonly denoted by $h_a, h_b, h_c$. We need to find the correct relationship between $(h_a + h_b + h_c)$ and $(a + b + c)$.

Analysing Altitudes in an Acute-Angled Triangle

An acute-angled triangle is a triangle where all three interior angles are less than 90 degrees. A key property of acute-angled triangles is that the foot of each altitude lies inside the opposite side (strictly between the two vertices of that side).

Consider the altitude AL from vertex A to the side BC. Since L lies between B and C, the altitude AL forms two right-angled triangles: $\triangle ABL$ (right-angled at L) and $\triangle ACL$ (right-angled at L).

• In the right-angled triangle $\triangle ABL$, AL is one leg and AB is the hypotenuse. In any right-angled triangle, the hypotenuse is the longest side. Therefore, the length of the leg AL must be less than the length of the hypotenuse AB.

This gives us the inequality: $AL < AB$, or $h_a < c$.

• Similarly, in the right-angled triangle $\triangle ACL$, AL is one leg and AC is the hypotenuse. Thus, the length of the leg AL must be less than the length of the hypotenuse AC.

This gives us the inequality: $AL < AC$, or $h_a < b$.

Applying the Principle to All Altitudes

We can apply the same reasoning to the other two altitudes:

• For the altitude BM from vertex B to side AC, M lies between A and C. BM forms right triangles $\triangle BMA$ (right-angled at M) and $\triangle BMC$ (right-angled at M).
  • In $\triangle BMA$, BM is a leg and AB is the hypotenuse. So, $BM < AB$, or $h_b < c$.
  • In $\triangle BMC$, BM is a leg and BC is the hypotenuse. So, $BM < BC$, or $h_b < a$.
• For the altitude CN from vertex C to side AB, N lies between A and B. CN forms right triangles $\triangle CNA$ (right-angled at N) and $\triangle CNB$ (right-angled at N).
  • In $\triangle CNA$, CN is a leg and AC is the hypotenuse. So, $CN < AC$, or $h_c < b$.
  • In $\triangle CNB$, CN is a leg and BC is the hypotenuse. So, $CN < BC$, or $h_c < a$.

Comparing Sum of Altitudes and Sum of Sides

Now we have several inequalities involving the altitude lengths and side lengths:

• $h_a < b$
• $h_b < c$
• $h_c < a$

Let's add these three inequalities together:

Rearranging the terms on the right side, we get:

Substituting the notation from the question:

This inequality shows that the sum of the lengths of the altitudes of an acute-angled triangle is less than the sum of the lengths of its sides.

Evaluating the Options

Let's compare our derived inequality with the given options:

1. $AL + BM + CN = AB + BC + CA$
2. $AL + BM + CN < AB + BC + CA$
3. $AL + BM + CN > AB + BC + CA$
4. $AL + BM = AB + BC$

Our result, $AL + BM + CN < AB + BC + CA$, matches option 2.

Conclusion

Based on the geometric properties of altitudes in an acute-angled triangle, the sum of the lengths of the altitudes is always less than the sum of the lengths of the sides.

Revision Table: Triangle Altitudes and Sides Inequality

Additional Information: Properties of Altitudes

Altitudes are important lines in a triangle. Here are a few more facts about them:

• The three altitudes of a triangle are concurrent, meaning they intersect at a single point.
• The point of intersection of the altitudes is called the orthocenter.
• In an acute-angled triangle, the orthocenter lies inside the triangle.
• In a right-angled triangle, the orthocenter is at the vertex with the right angle.
• In an obtuse-angled triangle, the orthocenter lies outside the triangle.
• The lengths of the altitudes are inversely proportional to the lengths of the corresponding sides. That is, $a h_a = b h_b = c h_c = 2 \times Area$. This means the longest altitude is perpendicular to the shortest side, and the shortest altitude is perpendicular to the longest side.