Fill in the black. in the triangle below , x=_ . round your answer to two decimal places.

x ≈ 44.80    

Step-by-step explanation:

As it's a right angle triangle so we will use rules of trigonometry.

The tangent of the angle = the length of the opposite side /

                                             the length of the adjacent side

First we need to find angle z

angle z

52° + z° + 90° = 180°       sum of angles of triangle

142° + z° = 180°

z° = 180 - 142    

z° = 38°

x

Tan z° = opp / adj

Tan 38° = 35 / x

x = 35 / Tan 38°

x = 35 / 0.78

x = 44.7979

x ≈ 44.80                         (correct to 2 decimal place)

Opposite angles in a cyclic quadrilateral add up to 180°

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