c To be able to use either equations, they must give the same value of mAB.
A
aExample Diagrams:
B
bEquation I: mAB = 2 m∠ BAC
Equation II: mAB =360^(∘)- 2 m∠ BAC
C
c m∠ BAC = 90^(∘)
Practice makes perfect
a We are asked to draw two diagrams to illustrate the given situation. We have been told that points A and B are on a circle and t is the tangent line containing A. Let's begin with this part!
Additionally, we have also been told that line t contains another point C. Since point C can be on either side of point A, there will be two diagrams.
Notice these are example diagrams. We can draw many other pairs of diagrams that illustrate the given situation.
b In this part we will write an equation for mAB in terms of m∠ BAC for each diagram. In each diagram, ∠ BAC is formed by the tangent line intersected by a chord. Therefore, let's first recall the Tangent and Intersected Chord Theorem (Theorem 10.14).
Tangent and Intersected Chord Theorem
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.
Considering this theorem, we will begin by writing an equation for the first diagram. Let's identify the intercepted arc of the first diagram.
Now that we can see the major arc AB is the intercepted arc of ∠ BAC, we can write the equation as follows.
m∠ BAC = 1/2 mAB
⇕
mAB = 2 m∠ BAC
Next, we will continue with the second diagram. Since we do not have a point on the circle other than A and B, we will label the intercepted arc as (360^(∘)-mAB) in the second diagram.
From here, we can write the second equation.
m∠ BAC = 1/2(360-mAB)
⇕
mAB = 360-2 m∠ BAC
c We want to find the measure of ∠ BAC for which we can use either equation from Part B to find mAB. To be able to use either of them, they must give the same value of mAB. Therefore, we will first write a system of equations.
