ABC is a triangle, where `angleBAC = 90^(@)`, If BC = 10 cm. What is the length of the median AD, in cm?

Options

A. 8
B. 5
C. 6
D. 4

Answer

To find the length of the median AD in triangle ABC where angle BAC = 90° and BC = 10 cm, we can follow these steps:

### Step 1: Understand the triangle configuration
Since triangle ABC is a right triangle with angle BAC = 90°, we can denote the points as follows:
- A is the vertex at the right angle.
- B and C are the other two vertices.

### Step 2: Identify the length of segment BC
We are given that the length of side BC = 10 cm.

### Step 3: Determine the midpoint D of side BC
The median AD connects vertex A to the midpoint D of side BC. To find the length of the median, we first need to find the length of BD and DC:
- Since D is the midpoint of BC, we have:
\[ BD = DC = \frac{BC}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

### Step 4: Use the median length formula
In a right triangle, the length of the median from the right angle to the hypotenuse can be calculated using the formula:
\[ AD = \frac{1}{2} \sqrt{2AB^2 + 2AC^2 - BC^2} \]
However, since we are not given the lengths of AB and AC, we can use a simpler property of medians in right triangles.

### Step 5: Use the property of the median in a right triangle
In a right triangle, the median to the hypotenuse is half the length of the hypotenuse. Since BC is the hypotenuse:
\[ AD = \frac{1}{2} \times BC = \frac{1}{2} \times 10 \text{ cm} = 5 \text{ cm} \]

### Conclusion
Thus, the length of the median AD is 5 cm.