Astudy of 50 high school students showed that exactly 25 of them took biology, exactly 20 of them took chemistry, and exactly 12 of them took both subjects. how many of the 50 students took neither biology nor chemistry?

The answer is 5 because if you add up the people who took Biology and Chemistry you get 45 then if you subtract that from 50 you get 5. You don't pay any attention to the 12 because it's only there to trick you. You don't need to know that exactly 12 people took both Biology and Chemistry bc they are already accounted for in the 25 and 20.

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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