Question

Please help I really need this answer since I am preparing for my nationals

Answer 1

Answer:   $\bold{a)\ \overline{CD}=6\qquad b)\ \overline{AD}=\dfrac{32}{3}\qquad c)\ \overline{AC}=\dfrac{50}{3}\qquad d)\ \overline{OB}=\dfrac{25}{3}}$

Step-by-step explanation:

a) First, use the Pythagorean Theorem to find CD:

8² + (CD)² = 10²     →     CD = 6

d) Next, find OB.  Notice that both OB and OC are the radius of the circle.

Using the Segment Addition postulate, we know the radius (r) = CD + DO  →  DO = r - 6

Use the Pythagorean Theorem for ΔOBD to find OB:

(DO)² + (BD)² = (OB)²

(r - 6)² + 8² = r²

r² - 12r + 36 + 64 = r²

   -12r + 100        = 0

             100        = 12r

              $\dfrac{25}{3}\quad = r$              

c) Now, find AC. Notice that AC is the diameter.

Diameter = 2r

               = 2(OB)

               $=2\bigg(\dfrac{25}{3}\bigg)\qquad =\dfrac{50}{3}$  

b) Lastly, find AD. Using the Segment Addition Postulate, we know that

AD + CD = AC   →   AD = AC - CD

                                      $= \dfrac{50}{3}-6\qquad \rightarrow \quad \dfrac{50}{3}-\dfrac{18}{3}\qquad =\dfrac{32}{3}$