1. In triangle ABC , AD, BE, CF are the altitudes and O is ortho-center. Which of the following triangles has the ortho-centre 'B'?

(a) triangle BOC

(b) triangle ADC

(c) triangle AOC

(d) triangle ADB

In a triangle, the orthocenter is the point where all three altitudes intersect. Now, let's consider the altitudes and the properties of the orthocenter in triangle ABC, with AD, BE, and CF as altitudes.

Given:
- \( O \) is the orthocenter of triangle ABC.
- We need to find which triangle among the options has \( B \) as its orthocenter.

#### Analysis:

1. **Orthocenter properties**:
   - In triangle ABC, \( O \) is the orthocenter.
   - When we look at triangle BOC (with vertices at B, O, and C), its orthocenter will correspond to the original vertex \( B \) of triangle ABC because the altitudes of triangle ABC pass through the orthocenter \( O \).

2. **Checking the options**:
   - For triangle BOC, the altitude from \( B \) in triangle ABC corresponds to an altitude in triangle BOC.
   - This gives us the orthocenter of triangle BOC as \( B \).

Hence, the triangle that has the orthocenter at \( B \) is:

### **Answer: (a) triangle BOC**

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How do you locate the orthocenter in any triangle?
2. What are the special properties of the orthocenter in an acute triangle?
3. How do the orthocenter and centroid differ in terms of their location in a triangle?
4. Can the orthocenter of a triangle lie outside the triangle? If yes, under what conditions?
5. What is the relationship between the orthocenter and circumcenter of a triangle?

**Tip:** The orthocenter lies inside the triangle if it's acute, on the vertex of a right triangle, and outside for an obtuse triangle.

In triangle ABC, AD, BE, CF are the altitudes and O is the orthocenter. Which of the following triangles has the orthocenter 'B'? (a) triangle BOC (b) triangle ADC (c) triangle AOC (d) triangle ADB

In this geometry problem, learn how to determine which triangle has its orthocenter at B. The problem involves understanding the orthocenter and properties of altitudes in a triangle, with a focus on triangle BOC. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation