Question

Given circle X with diameter VW and radius XT . TU is tangent to X at T. If TW =6 and VW =12 solve for TV. Round your answer to the nearest if necessary. If the answer cannot be determined click cannot be determined

Answer 1

We are given a circle with center X.

Other vital pieces of information given are;

`\begin{gathered} VW=\text{Diameter}=12 \\ XW=\text{Radius}=6 \\ XT=\text{Radius}=6 \\ TW=\text{Chord}=6 \end{gathered}`

We shall now extract triangle TWX and we'll have the following dimensions as indicated;

Note that we now have an equillateral triangle. Since all three sides are give as 6 units each (as indicated above), then all three sides will be 60 degrees each. All angles of an equilateral triangle are equal and all sum up 180 degrees.

To solve for side TV, we shall sketch a second triangle as shown below;

Observe that the diameter VW (12 units) is divided into two radii, which are;

`\begin{gathered} XV=6 \\ XW=6 \end{gathered}`

XT is also a radius which is 6. That gives us an isosceles triangle.

An isosceles triangke has two sides equal and the two base angles which are opposite the two equal sides are equal in measure.

Observe also that the line VW already has angle 60 degrees cut off (from triangle TWX). The remainder of the line would be 120 degrees.

"Angles on a straight line sum up to 180 degrees."

That way we can determine that angle

`\angle TXV=120\degree`

The other two angles would be;

`\begin{gathered} \angle T=\angle V=((180-120))/(2) \\ \angle T=\angle V=(60)/(2) \\ \angle T=\angle V=30\degree \end{gathered}`

That explains triangle TVX as sketched above.

To now solve for side TV, we shall apply the sine rule which is;

`(a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C)`

We can substitute the given values and we'll have;

`(6)/(\sin30)=(x)/(\sin 120)`

30 degrees is a special angle with a determined ratio and that is;

`\sin 30=(1)/(2)`

We can substitute for this into the equation as well as the value of sin 120 by use of a calculator;

`(6)/((1)/(2))=(x)/(0.866)`

We will refine the left side of the equation;

`\begin{gathered} (6)/(1)*(2)/(1)=(x)/(0.866) \\ 12=(x)/(0.866) \end{gathered}`

Next step, we cross multiply;

`\begin{gathered} 12*0.866=x \\ x=10.392 \end{gathered}`

Rounded to the nearest tenth;

ANSWER:

`x=10.4`