In the given figure, RQ is a tangent to the circle with centre O. If SQ = 6 cm and QR = 4 cm, then OR is equal to

The problem seems to be a geometry problem involving a circle and a tangent, but the figure is not provided. However, based on the given information, we can infer that this is a right triangle problem because a radius perpendicular to a tangent at the point of tangency forms a right angle.

Here are the steps to solve the problem:

1. Identify the right triangle. In this case, it's triangle ORQ, where O is the center of the circle, R is the point of tangency, and Q is another point on the tangent line.

2. Apply the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: a² + b² = c².

3. Substitute the given values into the equation. In triangle ORQ, RQ (which is the hypotenuse) is 4 cm, and SQ is 6 cm. Since RQ is a tangent to the circle at point R, and OR is a radius, the length of OQ is the sum of OR and RQ, or OR + 4 cm.

4. Solve for OR. The Pythagorean theorem equation becomes: (OR + 4 cm)² = (6 cm)² + (4 cm)². Simplifying this gives: OR² + 8OR + 16 = 36 + 16. Subtracting 16 from both sides gives: OR² + 8OR = 36. This is a quadratic equation in the form of ax² + bx + c = 0, where a = 1, b = 8, and c = -36.

5. Solve the quadratic equation. The solutions to the quadratic equation are given by the formula: x = [-b ± sqrt(b² - 4ac)] / (2a). Substituting the values of a, b, and c gives: OR = [-8 ± sqrt((8)² - 41(-36))] / (2*1) = [-8 ± sqrt(64 + 144)] / 2 = [-8 ± sqrt(208)] / 2 = [-8 ± 14.42] / 2.

6. Find the possible values of OR. The equation gives two possible values: OR = (-8 + 14.42) / 2 = 3.21 cm, or OR = (-8 - 14.42) / 2 = -11.21 cm.

7. Choose the valid solution. Since a length cannot be negative, the valid solution is OR = 3.21 cm.