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 find the length of the longer segment of the diameter, we can use the properties of a circle.
First, let's draw a diagram to visualize the problem.
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A
/|
/ |
/ |
_____/ |______
B C D
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In the diagram, consider a circle with center point C (the center of the circle). The diameter of the circle is AD, and the chord is AB. The shorter segment of the diameter is CB, which measures 4 inches. We want to find the length of the longer segment, CD.
Now, let's analyze the properties of a circle.
One important property is that a diameter of a circle intersects any chord that is perpendicular to it at the midpoint of the chord.
In our case, the diameter AD intersects the chord AB at point C, which is the midpoint of the chord AB.
Since C is the midpoint of AB, that means AC = CB = 4 inches.
Now, we can use the fact that the sum of the lengths of the two segments of a chord is equal to the length of the diameter.
In other words, AC + CD = AD.
We know that AC = 4 inches and AD is the diameter.
Therefore, AC + CD = 4 + CD = AD.
Since AD is the diameter, it is twice the length of CD.
So, 2CD = 4 + CD.
Subtracting CD from both sides gives: CD = 4 inches.
Therefore, the length of the longer segment of the diameter (CD) is 4 inches.