7.3 Q-9
Question Statement
Prove that the altitudes of a triangle are concurrent.
Background and Explanation
In geometry, the altitudes of a triangle are the perpendicular lines drawn from each vertex to the opposite side (or the line containing the opposite side). The point where all three altitudes of a triangle meet is called the orthocenter of the triangle. The theorem we need to prove states that the altitudes of any triangle are concurrent, meaning they all intersect at a single point, the orthocenter.
To prove this, we will use vector analysis and properties of perpendicularity.
Solution
Let the vertices of triangle be denoted by , , and . The goal is to show that the altitudes of this triangle are concurrent, meaning the altitudes meet at a common point, say .
Step 1: Define the altitudes and their relationships
We are given that and are the altitudes of the triangle, where:
- is the foot of the altitude from vertex onto side ,
- is the foot of the altitude from vertex onto side .
From the properties of altitudes, we know:
- ,
- .
We aim to show that the altitudes , , and are concurrent, meaning they all intersect at point .
Step 2: Perpendicularity and vector equations
Since , the vector from to (the point of intersection of the altitudes) must be perpendicular to vector . Therefore, we have the following equation:
This means that the dot product between and must be zero:
Similarly, for altitude , we know that . Hence, we have:
This leads to the equation:
Step 3: Combine the equations
Now, combining Equation 1 and Equation 2:
This can be rewritten as:
This equation indicates that the point , where the altitudes meet, satisfies the condition for perpendicularity.
Step 4: Conclusion
Since we have shown that the altitudes are perpendicular to the corresponding sides and meet at the same point, we conclude that the altitudes of the triangle are indeed concurrent at the orthocenter .
Thus, we have proved that the altitudes of a triangle are concurrent.
Key Formulas or Methods Used
- Dot product of vectors: For two vectors and , the dot product is given by:
Two vectors are perpendicular if their dot product is zero:
- Perpendicularity condition: If two vectors are perpendicular, their dot product must be zero:
Summary of Steps
- Define the altitudes of the triangle and , and their corresponding perpendicularity conditions.
- Use the vector dot product to express the condition of perpendicularity for the altitudes.
- Combine the equations for the perpendicularity of the altitudes to show that the altitudes meet at a common point.
- Conclude that the altitudes are concurrent, meaning they meet at the orthocenter .