Answer:
PC = 9.6 units (nearest tenth)
Step-by-step explanation:
Tangent
A straight line that touches a circle at only one point.
The tangent of a circle is always perpendicular to the radius.
Two-Tangent Theorem
If two tangents to a circle meet at one exterior point, the tangent segments are congruent.
Therefore:
- Tangent segments PC and QC are equal in length.
- AQ is the radius of the circle, and so triangle AQC is a right triangle with ∠AQC = 90°.
Find the measure of QC by using the cos trigonometric ratio.
Cos trigonometric ratio
[tex]\sf \cos(\theta)=\dfrac{A}{H}[/tex]
where:
- [tex]\theta[/tex] is the angle
- A is the side adjacent the angle
- H is the hypotenuse (the side opposite the right angle)
From inspection of the given diagram:
- [tex]\theta[/tex] = ∠ACQ = 29°
- A = QC
- H = AC = 11
Substitute the given values into the formula and solve for QC:
[tex]\implies \sf \cos(29^{\circ})=\dfrac{QC}{11}[/tex]
[tex]\implies \sf QC=11\cos(29^{\circ})[/tex]
[tex]\implies \sf QC=9.620816779...[/tex]
[tex]\implies \sf QC=9.6\:\:units\:(nearest\:tenth)[/tex]
As PC = QC then PC = 9.6 units (nearest tenth).
Learn more about tangents here:
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Learn more about trig ratios here:
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