Questions: Find the value of x.

Find the value of x.
Transcript text: Find the value of $x$.
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Solution

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Solution Steps

Step 1: Identify the Theorem

The problem involves a circle with a triangle inscribed in it. We can use the Power of a Point theorem, which states that for a point outside a circle, the product of the lengths of the segments of one secant line is equal to the product of the lengths of the segments of another secant line drawn from the same point.

Step 2: Apply the Power of a Point Theorem

In this problem, point C is outside the circle, and CE and CD are secant lines. According to the Power of a Point theorem: \[ CE \cdot CD = CB \cdot CA \]

Step 3: Substitute the Given Values

From the diagram:

  • \( CE = 4 \)
  • \( CD = 4 + 2x \) (since \( ED = 2x \))
  • \( CB = 8 \)
  • \( CA = 8 + 4 = 12 \)

Substitute these values into the equation: \[ 4 \cdot (4 + 2x) = 8 \cdot 12 \]

Step 4: Solve for x

Simplify and solve the equation: \[ 4 \cdot (4 + 2x) = 96 \] \[ 16 + 8x = 96 \] \[ 8x = 80 \] \[ x = 10 \]

Final Answer

The value of \( x \) is 10.

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