In two concentric circles, a chord of length 8cm of the larger circle touches the smaller circle . If the radius of the larger circle is 5cm then find the radius of the smaller circle.
touches ---> tangent ?
the radius of the smaller circle is perpendicular to the midpoint of the larger chord
the radius of the larger circle is the hypotenuse , and half of the chord is the longer side
use Pythagoras to find the radius of the smaller circle
hint: the triangle is a Pythagorean triple (all sides are integers)
To find the radius of the smaller circle, we can use the properties of tangents to circles.
First, let's draw a diagram to better visualize the problem.
We have two concentric circles. Let's label the center of the circles as O. The larger circle has a radius of 5cm, so we can label the point where the chord touches the smaller circle as A, and the point where the chord intersects the larger circle as B.
Since the chord AB is tangent to the smaller circle, we know that the radius of the smaller circle (OA) is perpendicular to the chord AB.
Next, let's use the property of tangents that tells us that a line that's tangent to a circle is perpendicular to the radius drawn to the point of tangency.
Since the chord AB is perpendicular to the radius OA, we can draw a right-angled triangle, with OA as the hypotenuse and AB as the side opposite the right angle.
The length of the chord AB is given as 8cm. In a right-angled triangle, the hypotenuse is the longest side, which in this case is OA. We are given the hypotenuse (OA) and the length of the opposite side (AB).
We can now apply Pythagoras' theorem to find the length of the radius OA.
Pythagoras' theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have:
OA^2 = AB^2 + OB^2
Substituting the values we know:
(5)^2 = (8)^2 + OB^2
25 = 64 + OB^2
OB^2 = 25 - 64
OB^2 = -39
We encountered a problem here. The equation has no real solution because the value under the square root is negative.
This means that there is no real solution for the radius of the smaller circle. The given information or problem statement might not be correct, or there might be an error in the values provided. Please double-check the question and the given values.