Given:
- A circle with center O
- ST = 3 cm

To find:
- Radius of the circle when RS ⊥ PQ

Solution:

Step 1: Understand the problem
- We are given a circle with center O and a line segment ST.
- We need to find the radius of the circle when RS ⊥ PQ.

Step 2: Analyze the figure
- Let's analyze the given figure to understand it better.
- The line segment ST is given, but the lengths of RS and PQ are not mentioned.
- We need to find the radius of the circle, which is the distance from the center O to any point on the circle.

Step 3: Identify the key points
- To find the radius, we need to identify the key points in the figure.
- The center of the circle is point O.
- The points R, S, P, and Q are on the circle.
- RS and PQ are perpendicular to each other.

Step 4: Use the properties of perpendicular lines
- Since RS ⊥ PQ, we can use the properties of perpendicular lines to find the radius of the circle.
- Perpendicular lines intersect at a right angle.
- The distance from the center of a circle to a tangent is equal to the radius of the circle.
- Therefore, if RS ⊥ PQ, then the distance from O to RS is equal to the radius of the circle.

Step 5: Find the radius
- Since RS ⊥ PQ, the distance from O to RS is equal to the radius of the circle.
- We can find the distance from O to RS by drawing a perpendicular from O to RS and measuring its length.
- Let's call the point where the perpendicular intersects RS as M.

Step 6: Use Pythagoras theorem
- To find the length of OM, we can use Pythagoras theorem since OSM is a right-angled triangle.
- In triangle OSM, OS² = OM² + SM².
- Since ST = 3 cm, SM = 3/2 cm (half of ST).
- Let's assume OM = x cm.

Step 7: Substitute the values and solve the equation
- Substituting the known values into the Pythagoras theorem equation, we get:
OS² = x² + (3/2)²
OS² = x² + 9/4
- Since O is the center of the circle, OS is equal to the radius of the circle.
- Let's assume the radius of the circle is R.

Step 8: Substitute the values and solve the equation
- Substituting the values into the equation, we get:
R² = x² + 9/4
- This equation gives the relationship between the radius of the circle and the length of the perpendicular from the center O to RS.

Step 9: Solve for the radius
- Since we know the length of ST is 3 cm, we can solve for the value of x using the Pythagoras theorem equation.
- Substituting x = 3/2 into the equation, we get:
R² = (3/2)² + 9

1.5 is the radius...