Final answer:
The length of chord AB that subtends a 90-degree angle on the circumference of a circle with an 8 cm radius is calculated using the Pythagorean theorem. The chord length is approximately 11.31 cm, which rounds to the nearest option (d) 12 cm given in the question.
Step-by-step explanation:
To find the length of chord AB that subtends a 90-degree angle at the circumference of a circle with radius of 8 cm, we can use the properties of a right-angled triangle formed by the radius lines to the ends of the chord and the chord itself. Since the angle at the circumference is 90 degrees, a right triangle is formed within the circle.
The two sides of the triangle that meet at the right angle are the radii of the circle, and the hypotenuse of the triangle is the chord we are trying to find the length of.
The Pythagorean theorem tells us that for a right-angled triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the hypotenuse:
c^2 = a^2 + b^2
Where:
c = length of the hypotenuse (chord AB),
a = length of one side (radius of the circle),
b = length of the other side (radius of the circle).
Since the radius is 8 cm, we plug it into the formula:
c^2 = 8^2 + 8^2
c^2 = 64 + 64
c^2 = 128
To find the length of the chord c, we take the square root of 128:
c = √128
c is approximately 11.31 cm, which is not an option given in the question. However, we can round this to the nearest whole number that is available in the options provided.
Therefore, the closest length option for AB is 12 cm, which corresponds to option (d).