Question

Let the three sides of a triangle are on the lines $4x - 7y + 10 = 0$, $x + y = 5$, and $7x + 4y = 15$. Then the distance of its orthocenter from the orthocenter of the triangle formed by the lines $x = 0$, $y = 0$, and $x + y = 1$ is

• A. 5
• B. $\sqrt{5}$
• C. $\sqrt{20}$
• D. 20

Hint:

The orthocenter of a triangle is the intersection of its altitudes.

Answer 1

The Correct Option is B

1. Identify the vertices of the triangle: 
- Solve the system of equations to find the intersection points of the lines. 
2. Find the orthocenter of the triangle: 
- The orthocenter is the intersection of the altitudes. 
3. Find the orthocenter of the triangle formed by $x = 0$, $y = 0$, and $x + y = 1$: 
- The orthocenter is the intersection of the altitudes. 
4. Calculate the distance between the two orthocenters: 
- Use the distance formula to find the distance between the two points. 
Therefore, the correct answer is (2) $\sqrt{5}$.