A diameter of a circle is perpendicular to a chord whose length is 12 inches. If the length of the shorter segment of the diameter is 4 inches, what is the length of the longer segment of the diameter?
To solve this problem, we can use the properties of perpendicular chords in a circle.
Let's label the shorter segment of the diameter as "x" inches and the longer segment as "y" inches.
According to the problem, the length of the shorter segment is 4 inches, so we have x = 4.
Since the diameter is perpendicular to the chord, we know that the segments of the diameter adjacent to the chord form a right triangle. The longer segment of the diameter will be the hypotenuse of this right triangle.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the shorter segment of the diameter (x) and the chord length (12 inches) are the other two sides of the right triangle.
Therefore, we have x^2 + y^2 = 12^2.
Substituting the value of x, we get 4^2 + y^2 = 12^2.
Simplifying the equation, we have 16 + y^2 = 144.
Subtracting 16 from both sides, we have y^2 = 128.
Taking the square root of both sides, we have y = sqrt(128).
Simplifying this, we get y ≈ 11.31 inches.
Therefore, the length of the longer segment of the diameter is approximately 11.31 inches.
To solve this problem, we need to apply the properties of a circle and use some geometry concepts.
Let's consider a circle with a perpendicular diameter and a chord.
Since the diameter is perpendicular to the chord, it means that the chord is bisected by the diameter. In other words, the diameter divides the chord into two equal parts.
According to the problem statement, the length of one segment of the diameter is 4 inches. Therefore, the other segment must also be 4 inches because they are equal due to the bisecting property.
Now, to find the length of the longer segment of the diameter, we need to add the lengths of both segments.
Length of shorter segment + Length of longer segment = Diameter
4 inches + Length of longer segment = Diameter
Since the diameter is the sum of the lengths of both segments, and both segments are equal at 4 inches, we can substitute "Diameter" with "8 inches" in the equation.
4 inches + Length of longer segment = 8 inches
Now, we can solve for the length of the longer segment by subtracting 4 inches from both sides of the equation:
Length of longer segment = 8 inches - 4 inches
Length of longer segment = 4 inches
So, the length of the longer segment of the diameter is 4 inches.