With < 3 the following two-column proof with missing statement proves that the diagonals of the rectangle bisect each other: statement reason abcd is a rectangle. given line segment ab and line segment cd are parallel | definition of a parallelogram line segment ad and line segment bc are parallel | definition of a parallelogram |alternate interior angles theorem line segment bc is congruent to line segment ad | definition of a parallelogram ∠adb ≅ ∠cbd | alternate interior angles theorem δade ≅ δcbe | angle-side-angle (asa) postulate line segment be is congruent to line segment de cpctc line segment ae is congruent to line segment ce cpctc line segment ac bisects line segment bd definition of a bisector which statement can be used to fill in the blank space? ∠abd ≅ ∠dbc ∠cad ≅ ∠acb ∠bda ≅ ∠bdc ∠cab ≅ ∠acb

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