In `Delta ABC, /_B = 90^@` and `BD _|_ AC`.
If CD = 10 cm and BD = 8 cm, find AD. 

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Triangle Configuration
We have triangle ABC where angle B is 90 degrees. BD is perpendicular to AC, creating two right triangles: ABD and BDC.

### Step 2: Label the Known Values
- CD = 10 cm
- BD = 8 cm
- Let AD = x cm (this is what we need to find)

### Step 3: Use the Properties of Right Triangles
In triangle ABD, we can express the tangent of angle θ (where θ is the angle at point A) as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AD}{BD} = \frac{x}{8} \]
(Equation 1)

In triangle BDC, we can express the tangent of angle θ as:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BD}{CD} = \frac{8}{10} \]
(Equation 2)

### Step 4: Set the Two Equations Equal
Since both expressions equal tan(θ), we can set them equal to each other:
\[ \frac{x}{8} = \frac{8}{10} \]

### Step 5: Solve for x
Cross-multiply to solve for x:
\[ 10x = 8 \times 8 \]
\[ 10x = 64 \]
\[ x = \frac{64}{10} \]
\[ x = 6.4 \text{ cm} \]

### Step 6: Conclusion
Thus, the length of AD is 6.4 cm.

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