From an external point P, a tangent PT has been drawn to a circle with centre at O and radius 3 cm, intersecting its concentric circle at A and B. If AB = 8 cm and OA = AP, the length PQ equals.
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Concentric circle problems with tangents often contain hidden 3-4-5 right-angled triangles!
Once you identify the perpendicular chord segment as 4 cm and the inner radius as 3 cm, you can immediately identify the outer radius as 5 cm, saving you calculation steps!
Step 1: Understanding the Question:
The topic of this question is Circles and concentric properties.
We are given two concentric circles with a common center \(O\).
The radius of the inner circle is \(OQ = 3\) cm.
A tangent is drawn from an external point \(P\) to touch the inner circle at the point of contact \(Q\).
This line intersects the outer circle at two points, \(A\) and \(B\), making \(AB\) a chord of the outer circle.
We are given \(AB = 8\) cm and \(OA = AP\). We need to calculate the length of \(PQ\).
Step 2: Key Formula or Approach:
- The radius to a point of tangency is perpendicular to the tangent line. Thus, \(OQ \perp PT\).
- A perpendicular dropped from the center of a circle to a chord bisects the chord. Since \(OQ \perp AB\), \(Q\) is the midpoint of chord \(AB\).
- Apply the Pythagoras theorem to the right-angled triangles:
\[ \Delta OAQ \implies OA^2 = OQ^2 + AQ^2 \]
\[ \Delta OQP \implies OP^2 = OQ^2 + PQ^2 \]
Step 3: Detailed Explanation:
• Determine the length of the bisected segments of chord \(AB\):
Since the radius \(OQ\) of the inner circle is perpendicular to the tangent line, it is also perpendicular to the chord \(AB\) of the outer circle.
Therefore, \(Q\) bisects the chord \(AB\):
\[ AQ = QB = \frac{AB}{2} = \frac{8}{2} = 4 \text{ cm} \]
• Use right-angled triangle \(\Delta OAQ\) to calculate the radius of the outer circle \(OA\):
By the Pythagoras theorem:
\[ OA^2 = OQ^2 + AQ^2 \]
Substitute the known values \(OQ = 3\) cm and \(AQ = 4\) cm:
\[ OA^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
\[ OA = \sqrt{25} = 5 \text{ cm} \]
So, the radius of the outer circle is 5 cm.
• Find the length of the segment \(AP\):
We are given that \(OA = AP\). Therefore:
\[ AP = 5 \text{ cm} \]
• Calculate the total length of the segment \(PQ\):
The point \(A\) lies on the line segment \(PQ\) between \(P\) and \(Q\).
Therefore:
\[ PQ = AP + AQ \]
Substitute the values \(AP = 5\) cm and \(AQ = 4\) cm into the equation:
\[ PQ = 5 + 4 = 9 \text{ cm} \]
Step 4: Final Answer:
The length of \(PQ\) is 9 cm.
Therefore, the correct option is (C).
