The sum of altitudes of a triangle is than the perimeter of the triangle._______

Options

A. greater
B. equal
C. half
D. less

Answer

To solve the problem, we need to show that the sum of the altitudes of a triangle is less than the perimeter of the triangle. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to prove that the sum of the altitudes of a triangle is less than the perimeter of the triangle.

**Hint:** Recall the definitions of altitudes and perimeter. The perimeter is the sum of all sides of the triangle.

### Step 2: Draw the Triangle
Let’s draw a triangle ABC. Label the vertices as A, B, and C.

**Hint:** Make sure to label the vertices clearly and draw the triangle accurately.

### Step 3: Draw the Altitudes
From each vertex of the triangle, draw a perpendicular line to the opposite side. Let:
- AD be the altitude from vertex A to side BC.
- BE be the altitude from vertex B to side AC.
- CF be the altitude from vertex C to side AB.

**Hint:** Remember that an altitude is a perpendicular segment from a vertex to the line containing the opposite side.

### Step 4: Apply the Triangle Inequality
For each triangle formed by the altitudes, we can apply the triangle inequality. For triangle ABD:
- The hypotenuse AB is greater than the altitude AD.
- Therefore, AB > AD.

For triangle ACD:
- The hypotenuse AC is greater than the altitude AD.
- Therefore, AC > AD.

**Hint:** The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step 5: Combine the Inequalities
From the above inequalities, we can write:
1. \( AB > AD \)
2. \( AC > AD \)

Adding these two inequalities gives:
\[ AB + AC > 2AD \]

**Hint:** When combining inequalities, ensure you are adding the correct sides.

### Step 6: Repeat for Other Altitudes
Now, apply the same reasoning for the other altitudes:
- For triangle ABE, we have \( AB + BC > 2BE \).
- For triangle ACF, we have \( AC + BC > 2CF \).

**Hint:** Write down these inequalities clearly as you derive them.

### Step 7: Sum All Inequalities
Now, we can sum all the inequalities:
\[ (AB + AC) + (AB + BC) + (AC + BC) > 2AD + 2BE + 2CF \]

This simplifies to:
\[ 2(AB + AC + BC) > 2(AD + BE + CF) \]

Dividing both sides by 2 gives:
\[ AB + AC + BC > AD + BE + CF \]

**Hint:** Make sure to simplify correctly when dividing by a constant.

### Step 8: Relate to Perimeter
The left side, \( AB + AC + BC \), is the perimeter of triangle ABC. Thus, we can write:
\[ \text{Perimeter} > \text{Sum of Altitudes} \]

### Conclusion
Therefore, we conclude that the sum of the altitudes of a triangle is less than the perimeter of the triangle:
\[ AD + BE + CF < AB + AC + BC \]

**Final Statement:** The sum of the altitudes of a triangle is less than the perimeter of the triangle.

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