Chords line PQ and line RS of a circle meet at X inside the circle. If RS = 38, PX = 6, and QX = 12, then what is the smallest possible value of RX?
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To find the smallest possible value of RX, we need to use a property of chords in a circle. The property states that if two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord.
Using this property, we can set up the following equation:
RX * SX = PX * QX
We are given that PX = 6 and QX = 12, so we can substitute these values into the equation:
RX * SX = 6 * 12
Since RX and SX represent the segments of chord RS, we can let a variable represent one of the segments, say RX = x. Then, SX = (38 - x) since RS = 38.
Therefore, the equation becomes:
x * (38 - x) = 6 * 12
x * (38 - x) = 72
Expanding the equation:
38x - x^2 = 72
Rearranging this equation:
x^2 - 38x + 72 = 0
To find the smallest possible value for RX, we need to find the minimum value of x, which occurs at the vertex of the parabolic equation. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In this equation, a = 1, b = -38, and c = 72. Plugging the values into the formula:
x = -(-38) / (2 * 1)
x = 38 / 2
x = 19
Therefore, the smallest possible value of RX is 19.