In the given figure, QA ⊥ AB and PB ⊥ AB. If AO = 20 cm, BO = 12 cm, PB = 18 cm, find AQ

Answer

To solve the problem, we will follow these steps:

1. **Identify the triangles**: We have two triangles, \( \triangle OAQ \) and \( \triangle OBP \). Since \( QA \) and \( PB \) are both perpendicular to \( AB \), we can use the properties of similar triangles.

2. **Establish similarity**: Since \( QA \perp AB \) and \( PB \perp AB \), we know that \( \angle OAQ = 90^\circ \) and \( \angle OBP = 90^\circ \). Additionally, \( \angle AOQ \) and \( \angle BOP \) are vertically opposite angles, which means they are equal. Therefore, by the Angle-Angle (AA) similarity criterion, \( \triangle OAQ \sim \triangle OBP \).

3. **Set up the proportion**: From the similarity of the triangles, we can set up the following proportion based on corresponding sides:
\[ \frac{AO}{BO} = \frac{AQ}{PB} \]
Substituting the known values:
\[ \frac{20 \, \text{cm}}{12 \, \text{cm}} = \frac{AQ}{18 \, \text{cm}} \]

4. **Cross-multiply to solve for \( AQ \)**:
\[ 20 \cdot 18 = 12 \cdot AQ \]
This simplifies to:
\[ 360 = 12 \cdot AQ \]

5. **Isolate \( AQ \)**:
\[ AQ = \frac{360}{12} \]
Simplifying this gives:
\[ AQ = 30 \, \text{cm} \]

Thus, the length of \( AQ \) is \( 30 \, \text{cm} \).