In a circle, the diameter AB is extended to an external point ... | Mathematics MCQ | Examoogle

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In a circle, the diameter AB is extended to an external point P. Tangent PT is drawn to point T on the circle. If PT = 12 cm and BP = 8 cm, then find the radius of the circle.

Correct Answer

📐 MA • Math Tangent Secant TheoremMathematicsGeometry

Option A

Quick Summary:

Use the property that the power of point P is $PT^2 = BP \times AP$ . Calculate $AP = 144 / 8 = 18$ , then $AB = AP - BP = 18 - 8 = 10$ . Radius is half of $AB$ , which is 5.

📐MAMath SolutionTangent Secant Theorem

📋 Given

Tangent length PT = 12 cm, External segment BP = 8 cm. The line segment AB is the diameter of the circle.

🔢 Formula Used

PT2=BP×APPT^2 = BP \times APPT2=BP×AP

⚡ Exam Hall Shortcut / Speed Trick

Use the property that the power of point P is $PT^2 = BP \times AP$ . Calculate $AP = 144 / 8 = 18$ , then $AB = AP - BP = 18 - 8 = 10$ . Radius is half of $AB$ , which is 5.

⚠️ Common Student Trap / Pitfall

Students often use $AP = 8$ or confuse $AP$ with the radius, forgetting that $AP$ is the total distance from the external point to the far end of the diameter ( $BP + AB$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

State the Tangent-Secant Theorem

According to the tangent-secant theorem, the square of the tangent is equal to the product of the external secant segment and the entire secant segment.

PT2=BP×APPT^2 = BP \times APPT2=BP×AP

Calculate the total secant length AP

Substitute the given values $PT = 12$ and $BP = 8$ into the formula to find the total length of the secant segment $AP$ .

122=8×AP  ⟹  144=8×AP  ⟹  AP=1448=1812^2 = 8 \times AP \implies 144 = 8 \times AP \implies AP = \frac{144}{8} = 18122=8×AP⟹144=8×AP⟹AP=8144​=18

Find the diameter AB

Since $AP = AB + BP$ , subtract $BP$ from $AP$ to find the diameter $AB$ .

AB=AP−BP=18−8=10AB = AP - BP = 18 - 8 = 10AB=AP−BP=18−8=10

Calculate the radius r

The radius $r$ of the circle is half of the diameter $AB$ .

r=AB2=102=5r = \frac{AB}{2} = \frac{10}{2} = 5r=2AB​=210​=5

✅

A is correct because the calculated radius of the circle is 5 cm.

Core Concepts Used

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Tangent-Secant Theorem Circle Properties Geometric Mean Theorem

💡 EXAM TIP

This theorem is a specific case of the Power of a Point theorem. It is frequently tested in geometry problems involving cyclic quadrilaterals and secants drawn from external points.

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