16 teams enter a competition. They are divided up into four Pools (A, B, C and D) of four teams each. Every team plays one match against the other teams in its pool.
After the Pool matches are completed:
the winner of Pool A plays the second placed team of Pool B
• the winner of Pool B plays the second placed team of Pool A
the winner of Pool C plays the second placed team of Pool D
• the winner of Pool D plays the second placed team of Pool C
The winners of these four matches then play semi-finals, and the winners of the semi-finals play in the final.
How many matches are played altogether?
63
32
23
31

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

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