b Mark the center of the given circle and connect it to the endpoints of the present chord. Then, use the Law of Cosines to write an equation for x.
A
a x ≈ 2.12
B
b x ≈ 10.88cm
Practice makes perfect
a Consider the given diagram.
We want to find the length x. In order to do so, we should first find the radius of the C from the diagram.
Radius
We know that m PMQ = 314^(∘). Since this angle, together with ∠ PCQ form a full angle, the measures of these angles have to sum to 360^(∘).
m PMQ + m ∠ PCQ = 360^(∘)Substituting the known value of m PMQ allows us to find the measure m ∠ PCQ
Now let's add the found measure to the diagram and let's focus on △ CRP. It is a right triangle, since PR is tangent to C at P.
Since we want to find the radius of C, we want to find the length CP. Notice that since we know m∠ PCR and length of the side opposite this angle, PR, we can use the tangent ratio to find CP.
tan ( m∠ PCR ) = PR/CP
Let's substitute 46^(∘) for m ∠ PCR and 5 for PR and solve the equation above for CP.
Let's again focus on the △ CRP. Notice that we know both legs of this triangle. Therefore, we can use the Pythagorean Theorem in order to find the hypotenuse. Let's do so!
Finally, notice that QR of length x is part of the larger segment CR. Also, since point Q lies on the circle, its distance from point C is equal to the radius. Therefore, we can use the Segment Addition Postulate to find the length x.
The lengths of the two smaller segments must sum to CR = 6.95.
4.83 + x ≈ 6.95 ⇒ x ≈ 2.12
b We want to find the value of x using the fact that the radius of the following circle is 7 centimeters.
Let's denote the endpoints of the chord from the diagram as A and B and mark the center of the circle and call it C. Next, connect C with A and B. Since the radius of a circle is 7 centimeters, both AC and BC are 7 centimeters long.
Now, let's use the Law of Cosines to relate AB to AC and BC, and the measure of ∠ ACB.
AB^2 = AC^2 + BC^2 - 2AC * BC * cos ∠ ACB
We know that AC = 7 centimeters and BC = 7 centimeters. Also, since m AB = 102^(∘), we have that the central angle ∠ ACB measures 102^(∘). Let's substitute these values into the above formula to get an equation for AB. We can also substitute x for AB to immediately get an equation for x. Let's do it!
