Answer:
x = 6
Explanation:
Chord Theorem
In a circle if two chords intersected at a point inside it there are four segments created, two in each cord, the products of the lengths of the line segments on each chord are equal
∵ AC and BD are two chords in a circle
∵ AC and BD intersect each other at a point inside the circle
- That means the products of the segments in each chord
are equal
∵ The segments of AC are (2x -
) and 4
∵ The segments of BD are x and (x - 4)
∴ (2x -
) × 4 = x × (x - 4)
- Simplify each side
∴ (2x)(4) - (
)(4) = (x)(x) - (x)(4)
∴ 8x - 36 = x² - 4x
- Subtract 4x from both sides
∴ - 36 = x² - 12x
- Add 36 to both sides
∴ 0 = x² - 12x + 36
- Switch the two sides
∴ x² - 12x + 36 = 0
Now let us factorize the left hand side into two factors
∵ x² = (x)(x)
∵ 36 = (-6)(-6)
∴ (x)(-6) + (x)(-6) = -6x + -6x = -12x ⇒ middle term
- That means the two factors are (x - 6) and (x - 6)
∴ The factors of x² - 12x + 36 are (x - 6) and (x - 6)
∴ (x - 6)(x - 6) = 0
- Equate the factor by 0
∵ x - 6 = 0
- Add 6 to both sides
∴ x = 6