. Determine whether a tangent line to circle O is shown in the diagram, for AB = 7.75, OB = 4, and AO = 8.75. Explain your reasoning. (The f

tangent condition: $$7.75^2+4^2=8.75^2$$

$$7.75^2+4^2\stackrel{?}{=}8.75^2$$

$$(\frac{31}{4})^2
+(4)^2\stackrel{?}{=}(\frac{35}{4})^2
\quad | \quad *4^2$$

$$31^2+(4^2)^2
\stackrel{?}{=}35^2$$

$$31^2+16^2
\stackrel{?}{=}35^2$$

$$961+256 \stackrel{?}{=}1225$$

$$1217\neq1225$$   not a tangent!

If it is a tangent then angle B is a right angle.  The easy way to test is with the pythagorean theorem.

7.75^2+4^2=8.75^2

76.0625=76.5625

This means that Angle b is not a right angle so Segment AB is not a tangent (though it is extraordinarily close to being one.)

IF AB is a tangent......then

AB^2 + BO^2 = AO^2  (because OB would have to meet AB at right angles)  .......therefore......

(7.75)^2 + (4)^2  =  (8.75)^2   ????

60.0625 + 16 = 76.5625   ???

76.0625 = 76.5625  ???      Nope......  AB is not a tangent!!!