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 relationship between the median, the base of the triangle, and the properties of the triangle itself. Here’s a step-by-step solution:

### Step 1: Understand the Median
The median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In this case, we are considering a triangle ABC with base BC and median AD drawn from vertex A to the midpoint D of side BC.

**Hint:** Remember that the median divides the opposite side into two equal segments.

### Step 2: Set Up the Given Information
According to the problem, the median AD is half of the base BC. This means we can express this relationship mathematically:
\[ AD = \frac{1}{2} BC \]

**Hint:** Use variables to represent the lengths of the sides for clarity.

### Step 3: Analyze the Triangle
Since D is the midpoint of BC, we can denote:
\[ BD = DC = \frac{1}{2} BC \]

Now, we have:
- \( AD = \frac{1}{2} BC \)
- \( BD = DC = \frac{1}{2} BC \)

**Hint:** Visualize the triangle and the median to understand how the segments relate to each other.

### Step 4: Apply the Properties of Triangles
In triangle ABD, we have:
- \( AB = AD \) (since AD is the median)
- \( BD = DC \)

This means triangle ABD is isosceles because two sides (AB and AD) are equal.

**Hint:** Recall the properties of isosceles triangles, particularly that the angles opposite the equal sides are also equal.

### Step 5: Establish Angles
Let the angles at A be denoted as:
- \( \angle ABD = \theta \)
- \( \angle DAB = \theta \)

Since triangle ABD is isosceles, we have:
\[ \angle ABD = \angle DAB \]

**Hint:** Use the properties of angles in triangles to find relationships.

### Step 6: Analyze Triangle ADC
Similarly, in triangle ADC:
- \( AD = DC \)
- Therefore, triangle ADC is also isosceles.

Let:
- \( \angle ACD = \alpha \)
- \( \angle DAC = \alpha \)

Thus:
\[ \angle ACD = \angle DAC \]

**Hint:** Look for symmetry in the triangle to identify equal angles.

### Step 7: Sum of Angles in Triangle ABC
The sum of angles in triangle ABC is:
\[ \angle BAC + \angle ABC + \angle ACB = 180^\circ \]

Substituting the angles we have:
\[ \alpha + \theta + \alpha + \theta = 180^\circ \]
\[ 2\alpha + 2\theta = 180^\circ \]
\[ \alpha + \theta = 90^\circ \]

**Hint:** Remember that the angles in a triangle always sum up to 180 degrees.

### Step 8: Conclusion
Since \( \angle BAC = 90^\circ \), triangle ABC is a right triangle.

Thus, the triangle will be a **right-angled triangle**.

**Final Answer:** The triangle will be a right-angled triangle.