In a `DeltaABC, D and E `are points on AC and BC respectively, AB and DE are perpendicular to BC. If `AB = 9 cm, DE = 3 cm and AC = 24 cm`, then AD is :

Options

A. 32cm
B. 16cm
C. 8 cm
D. 4 cm

Answer

To solve for the length of AD in triangle ABC, we can use the properties of similar triangles. Here’s a step-by-step solution:

### Step 1: Understand the triangle and the given values
We are given:
- AB = 9 cm
- DE = 3 cm
- AC = 24 cm

Points D and E are on sides AC and BC respectively, and both AB and DE are perpendicular to BC.

### Step 2: Set up the relationship using similar triangles
Since triangles ABD and CDE are similar (because they share angle B and both have right angles), we can set up a proportion based on the lengths of corresponding sides.

The proportion can be set up as follows:
\[ \frac{DE}{AB} = \frac{DC}{AC} \]

### Step 3: Substitute the known values into the proportion
Substituting the known values into the proportion:
\[ \frac{3}{9} = \frac{DC}{24} \]

### Step 4: Simplify the left side of the equation
Simplifying \(\frac{3}{9}\):
\[ \frac{1}{3} = \frac{DC}{24} \]

### Step 5: Cross-multiply to solve for DC
Cross-multiplying gives us:
\[ 1 \cdot 24 = 3 \cdot DC \]
\[ 24 = 3 \cdot DC \]

### Step 6: Solve for DC
Dividing both sides by 3:
\[ DC = \frac{24}{3} = 8 \text{ cm} \]

### Step 7: Find AD
Now that we have DC, we can find AD using the relationship:
\[ AD + DC = AC \]
Substituting the known values:
\[ AD + 8 = 24 \]
Solving for AD:
\[ AD = 24 - 8 = 16 \text{ cm} \]

### Final Answer
Thus, the length of AD is **16 cm**.

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