Final answer:
The radius of the circle is found using the Pythagorean Theorem. For the given right triangle with a tangent of 1 cm and hypotenuse of 17 cm, the radius is calculated to be 16 cm. Option D is correct.
Step-by-step explanation:
The question asks for the radius of a circle where the length of the tangent from an external point P to the circle is 1 cm, and the distance from P to the center of the circle is 17 cm. To find the radius of the circle, we use the Pythagorean Theorem because a tangent to a circle forms a right angle with the radius at the point of tangency.
We have a right triangle with one leg as the tangent (1 cm), the other leg as the radius of the circle (r), and the hypotenuse as the distance from the external point to the center (17 cm).
According to the Pythagorean Theorem, the sum of the squares of the legs equals the square of the hypotenuse:
r^2 + 1^2 = 17^2
r^2 + 1 = 289
r^2 = 288
r = sqrt(288)
r = 16 cm
Therefore, the correct answer is d) 16 cm.
The formula to find the length of a tangent from a point to a circle is:
Length of tangent = √(Distance of point from center² - Radius²)
In this case, the distance of point P from the center is 17 cm and the length of the tangent is 1 cm. Let's substitute these values into the formula:
1 = √(17² - Radius²)
Squaring both sides of the equation, we get:
1² = (17² - Radius²)
1 = 289 - Radius²
Rearranging the equation, we have:
Radius² = 289 - 1
Radius² = 288
Taking the square root of both sides, we find:
Radius = √288
Using a calculator, we can find the approximate value of the radius as 16 cm.