If the median drawn on the base of a triangle is half its base, the triangle will be:

Options

A. right-angled
B. acute-angled
C. obtuse-angled
D. equilateral

Answer

To solve the problem, we need to analyze the properties of a triangle when the median drawn to the base is half the length of the base.

### Step-by-Step Solution:

1. **Understanding the Median**:
The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In this case, we are considering a triangle ABC with base AC and vertex B.

2. **Setting Up the Triangle**:
Let the length of base AC be denoted as \( b \). According to the problem, the median \( BD \) (where D is the midpoint of AC) is half of the base. Therefore, we can express this as:
\[ BD = \frac{1}{2}b \]

3. **Using the Properties of a Triangle**:
In a triangle, if the median to one side is half the length of that side, we can use the properties of triangles to analyze the situation.

4. **Applying the Median Length Formula**:
The length of the median \( m_a \) from vertex A to side BC can be calculated using the formula:
\[ m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]
where \( a \) is the length of side BC, \( b \) is the length of side AC, and \( c \) is the length of side AB.

5. **Setting up the Equation**:
In our case, since \( BD = \frac{1}{2}b \), we can set up the equation:
\[ \frac{1}{2}b = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2} \]
By squaring both sides, we can eliminate the square root:
\[ \left(\frac{1}{2}b\right)^2 = \frac{1}{4}(2b^2 + 2c^2 - a^2) \]

6. **Simplifying the Equation**:
This simplifies to:
\[ \frac{1}{4}b^2 = \frac{1}{4}(2b^2 + 2c^2 - a^2) \]
Multiplying through by 4 gives:
\[ b^2 = 2b^2 + 2c^2 - a^2 \]
Rearranging yields:
\[ a^2 + b^2 = 2c^2 \]

7. **Identifying the Type of Triangle**:
The equation \( a^2 + b^2 = 2c^2 \) indicates that triangle ABC is a right triangle, where c is the hypotenuse. This is because it resembles the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

### Conclusion:
Thus, if the median drawn on the base of a triangle is half its base, the triangle will be a right triangle.