In a circle, a chord of length 5√2 centimetres subtends a right angle at the centre. Find the length of the radius of the circle in centimetres.

Introduction / Context:
This problem involves a circle, a chord and the central angle subtended by that chord. It tests your understanding of the relationship between chord length, radius and central angle. Recognising how to connect these elements using trigonometry or right triangle geometry is a valuable skill in circle based questions.

Given Data / Assumptions:

• There is a circle with centre O and radius R.
• A chord of the circle has length 5√2 centimetres.
• This chord subtends a right angle (90 degrees) at the centre O.
• We must find the radius R in centimetres.
• Standard Euclidean geometry and basic trigonometry can be used.

Concept / Approach:
The central angle at O is 90 degrees. The endpoints of the chord and the centre form an isosceles triangle with two sides of length R (the radii) and base equal to the chord. Because the central angle is 90 degrees, this triangle is also a right angled isosceles triangle. Using properties of right triangles, the chord length is related to the radius by a simple trigonometric formula: chord = 2 * R * sin(theta / 2). For theta = 90 degrees, sin(45 degrees) = √2 / 2, so chord = R * √2. We can then solve for R.

Step-by-Step Solution:
Step 1: Let the radius of the circle be R. Step 2: The chord subtends a central angle of 90 degrees at the centre O. Step 3: Use the chord length formula: chord length = 2 * R * sin(theta / 2). Step 4: Here theta = 90 degrees, so theta / 2 = 45 degrees. Step 5: sin(45 degrees) = √2 / 2. Step 6: Substitute into the formula: chord = 2 * R * (√2 / 2) = R * √2. Step 7: The chord is given as 5√2 cm, so R * √2 = 5√2. Step 8: Divide both sides by √2: R = 5 cm.

Verification / Alternative check:
Alternatively, you can think in terms of the right triangle formed by the radius lines and half the chord. If the chord length is 5√2, then half the chord is (5√2) / 2. The triangle with sides R, R and 5√2 has a right angle at the centre. Using Pythagoras in a right isosceles triangle, the hypotenuse (diameter) is 2R and each leg (radius) is R. The base (chord) corresponds to R√2. Setting R√2 = 5√2 again gives R = 5 cm, confirming the answer.

Why Other Options Are Wrong:
2.5 cm and 3.5 cm: These radii are too small, producing chords shorter than 5√2 cm for a 90 degree central angle.
7.5 cm and 10 cm: These are too large, generating chord lengths greater than 5√2 cm for the same central angle.

Common Pitfalls:
Some learners attempt to apply Pythagoras theorem directly without a clear diagram and mix up the sides of the triangle. Others may incorrectly use the full diameter instead of the radius in formulas. A careful sketch and the use of the standard chord formula or properties of a 45-45-90 triangle help make the relationships clear and reduce errors.

Final Answer:
The radius of the circle is 5 cm.