In a triangle ABC, if AB = 20 cm, AC = 21 cm and BC = 29 cm, then find the distance vertex A and mid point of BC.

Answer

To find the distance from vertex A to the midpoint D of side BC in triangle ABC where AB = 20 cm, AC = 21 cm, and BC = 29 cm, we can follow these steps:

### Step 1: Identify the triangle type
We start by checking if triangle ABC is a right triangle. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

### Step 2: Apply the Pythagorean theorem
Let's check if BC is the hypotenuse:
- BC² = AB² + AC²
- 29² = 20² + 21²
- 841 = 400 + 441
- 841 = 841

Since both sides are equal, triangle ABC is a right triangle with angle A being 90 degrees.

### Step 3: Find the midpoint D of side BC
The length of BC is 29 cm. The midpoint D divides BC into two equal parts:
- BD = DC = BC / 2 = 29 cm / 2 = 14.5 cm

### Step 4: Use trigonometry to find AD
Now, we can find the length of AD (the distance from vertex A to midpoint D). In triangle ADC, angle A is 90 degrees, and we can use the properties of a right triangle.

Using the tangent function:
- tan(45°) = opposite / adjacent
- Since angle A is 90 degrees and angle D is 45 degrees, we can say:
- tan(45°) = AD / CD
- 1 = AD / 14.5
- Therefore, AD = 14.5 cm

### Final Answer
The distance from vertex A to the midpoint D of side BC is **14.5 cm**.

---