In `DeltaABC`, the height CD intersects AB at D. The mid-points of AB and BC are P and Q respectively. If AD = 8 cm and CD = 6 cm, then the length of PQ is

Options

A. 3 cm
B. 7 cm
C. 9 cm
D. 5 cm

Answer

To solve the problem step by step, we will follow the given information and apply the necessary geometric principles.

### Step 1: Identify the triangle and given dimensions
We have triangle ABC with height CD intersecting AB at point D. We know:
- AD = 8 cm
- CD = 6 cm

### Step 2: Use the Pythagorean theorem to find AC
Since CD is the height from point C to line AB, triangle ACD is a right triangle. We can use the Pythagorean theorem to find the length of AC.

According to the Pythagorean theorem:
\[ AC^2 = AD^2 + CD^2 \]

Substituting the given values:
\[ AC^2 = 8^2 + 6^2 \]
\[ AC^2 = 64 + 36 \]
\[ AC^2 = 100 \]

Taking the square root of both sides:
\[ AC = \sqrt{100} = 10 \, \text{cm} \]

### Step 3: Find the midpoints P and Q
- P is the midpoint of AB.
- Q is the midpoint of BC.

### Step 4: Apply the midpoint theorem
According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.

Thus, since PQ is the line segment joining midpoints P and Q:
\[ PQ \parallel AC \]
\[ PQ = \frac{1}{2} \times AC \]

### Step 5: Calculate the length of PQ
Substituting the length of AC:
\[ PQ = \frac{1}{2} \times 10 \]
\[ PQ = 5 \, \text{cm} \]

### Conclusion
The length of PQ is 5 cm.

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