If angle bisector of a triangle bisect the opposite side, then what type of triangle is it?

Options

A. Right angled
B. Scalene
C. Similar
D. Isosceles

Answer

To determine the type of triangle when the angle bisector of a triangle bisects the opposite side, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Triangle and Angle Bisector**:
- Let's consider a triangle ABC, where we have vertices A, B, and C.
- The angle bisector is a line that divides an angle into two equal angles. For example, if we draw the angle bisector from vertex A to the opposite side BC, it will meet BC at point D.

2. **Applying the Angle Bisector Theorem**:
- According to the Angle Bisector Theorem, the angle bisector divides the opposite side (BC) in the ratio of the other two sides (AB and AC).
- This means that if AD is the angle bisector, then:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]

3. **Condition of the Opposite Side Being Bisected**:
- The question states that the angle bisector bisects the opposite side. This implies that BD = DC.
- Therefore, we can say that:
\[ BD = DC \implies \frac{BD}{DC} = 1 \]

4. **Setting Up the Ratios**:
- From the Angle Bisector Theorem, since BD = DC, we have:
\[ \frac{AB}{AC} = 1 \]
- This indicates that AB = AC.

5. **Conclusion**:
- Since two sides of triangle ABC (AB and AC) are equal, triangle ABC is an isosceles triangle.
- Additionally, if all three sides are equal (AB = AC = BC), then it can also be an equilateral triangle.

### Final Answer:
If the angle bisector of a triangle bisects the opposite side, then the triangle is an **isosceles triangle**. It can also be an equilateral triangle if all sides are equal.

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