Two secant segments are drawn to a circle from a point outside the circle. The external segment of the first secant segment is 8 centimeters and its internal segment is 6 centimeters. If the entire length of the second secant segment is 28 scentimeters, what is the length of its external segment?
To find the length of the external segment of the second secant segment, we can use a property of secants and tangents.
The property states that if two secant segments are drawn to a circle from the same external point, the product of the lengths of the external segment and its entire length is equal to the product of the lengths of the internal segment and its entire length.
Let's denote the external segment of the second secant segment as x.
According to the property, we can set up the following equation:
8 cm * (8 cm + x) = 6 cm * 28 cm
Now, let's solve for x:
64 cm^2 + 8x cm = 168 cm^2
Subtracting 64 cm^2 from both sides, we get:
8x cm = 104 cm^2
Dividing both sides by 8 cm, we find:
x = 13 cm
Therefore, the length of the external segment of the second secant segment is 13 centimeters.
To find the length of the external segment of the second secant segment, we need to use a theorem called the Intersecting Secants Theorem, which states that when two secant segments intersect outside a circle, the product of the lengths of the external segment of one secant and its full length is equal to the product of the lengths of the external segment of the other secant and its full length.
Let's assign variables to the given lengths:
Length of the external segment of the first secant segment = a (given as 8 centimeters)
Length of the internal segment of the first secant segment = b (given as 6 centimeters)
Length of the full second secant segment = x (given as 28 centimeters)
Length of the external segment of the second secant segment = y (to be determined)
The Intersecting Secants Theorem can be written as a * (a + b) = x * (x + y)
Substituting the given values into the equation, we have:
8 * (8 + 6) = 28 * (28 + y)
Simplifying the equation:
8 * 14 = 28 * (28 + y)
112 = 28 * (28 + y)
Divide both sides of the equation by 28:
112/28 = 28 * (28 + y) / 28
4 = 28 + y
Now, isolate y by subtracting 28 from both sides:
4 - 28 = 28 + y - 28
-24 = y
Therefore, the length of the external segment of the second secant segment is -24 centimeters. However, since length cannot be negative, there might be an error in the given information or the problem itself.