Skip to main content
Login
CSEEL - Centre for Scientific Exploration & Experiential Learning

Equality of Tangents Drawn from an External Point to a Circle

Experiment video

What you will learn

Verify that the lengths of tangents drawn from an external point to a circle are equal (Theorem 10.2) through paper folding, Euclidean geometric construction, and RHS congruency.

Materials needed

  • Drawing sheet (white chart paper)
    Drawing sheet (white chart paper)
    ₹50
    What is it?

    Tangent construction geometric acrylic kit with pre-drilled pivot holes and tangent rulers.

    Specifications
    Base Plate
    250 mm × 250 mm engraved millimeter grid board
    Rods
    Two slotted acrylic tangent arms with pivot pins
    Accuracy
    ± 0.1 mm mechanical tolerances
  • Geometry Box (Compass, 30 cm Metric Ruler, Protractor)
    NS
    Geometry Box (Compass, 30 cm Metric Ruler, Protractor)
    ₹95
    What is it?

    Heavy-duty steel drafting compass and ruling pens.

    Specifications
    Compass Length
    160 mm with extension arm
    Divider
    8-inch proportional divider
  • Tracing paper and colored pencils set
    Tracing paper and colored pencils set
    ₹140
    What is it?

    Precision steel ruler with metric 0.5 mm divisions.

    Specifications
    Length
    300 mm stainless steel rule
    Graduation
    0.5 mm and 1.0 mm etched scales
  • Fine craft paper-creasing tool
    Fine craft paper-creasing tool
    ₹185
    What is it?

    Standard laboratory apparatus used in Equality of Tangents Drawn from an External Point to a Circle.

    Specifications
    Category
    Mathematics Apparatus
    Grade
    CBSE / NEP 2020 Laboratory Standard

Tick items as you collect them. Tap + to see the product, specifications and buy options.

Stay safe

  • Compass pivot must be sharp and tightened to prevent slipping during arc construction.
  • The tangent must touch the circle at exactly one point (radius OT is strictly perpendicular to PT: ∠OTP = 90°).
  • Align the ruler with hairline accuracy along the points of tangency.
  • Crease paper cleanly along axis of symmetry OP for folding verification.

Steps

0 of 7 steps done
  • 1Draw a circle of radius r = 5 cm with center O on white chart paper.
    Observation

    Draw a circle of radius r = 5.0 cm with center O on white drawing board; mark external point P at OP = 13.0 cm.

    Safety

    Ensure straight line OP is measured accurately with steel rule.

  • 2Choose a point P outside the circle such that distance OP = 13 cm.
    Observation

    Find midpoint M of line OP by constructing its perpendicular bisector using compass arcs.

    Safety

    Open compass to radius > half of OP to draw intersecting bisector arcs.

  • 3Join OP and draw the perpendicular bisector of OP to find midpoint M.
    Observation

    With center M and radius MP = MO = 6.5 cm, draw a circle intersecting original circle at points T1 and T2.

    Safety

    Anchor compass point firmly at M to prevent center slip.

  • 4With M as center and radius MO = MP, draw a second circle intersecting the original circle at points T1 and T2.
    Observation

    Join PT1 and PT2; draw radii OT1 and OT2; verify angles ∠OT1P and ∠OT2P are exactly 90.0° using set square.

    Safety

    A tangent to a circle is perpendicular to the radius through the point of contact.

  • 5Join PT1 and PT2: these are the two tangents from external point P to the circle.
    Observation

    Measure tangent lengths PT1 and PT2 with vernier / divider; both measure exactly 12.00 cm.

    Safety

    Use fine dividers to transfer lengths directly onto steel rule scale.

  • 6Measure lengths PT1 and PT2 with a precision metric ruler (both measure exactly 12.0 cm: √(13² - 5²) = √(169 - 25) = √144 = 12 cm).
    Observation

    Cut along PT1, PT2 and fold along OP: triangle ΔOPT1 superimposes perfectly onto ΔOPT2, confirming equality.

    Safety

    Fold precisely along axis of symmetry OP.

  • 7Fold the paper along line OP: observe that ray PT1 superimposes precisely on PT2, verifying equality.

Tap + on a step to see what you will observe and any safety tips.

Record your observations

Construction TrialCircle Radius r (cm)Distance OP from Center (cm)Calculated Tangent √(OP²-r²) (cm)Measured PT1 (cm)Measured PT2 (cm)Difference |PT1 - PT2| (cm)
Trial 15.0 cm13.0 cm12.00 cm12.00 cm12.00 cm0.00 cm (Exact equality)
Trial 23.0 cm5.0 cm4.00 cm4.00 cm4.00 cm0.00 cm (Exact equality)
Trial 36.0 cm10.0 cm8.00 cm8.00 cm8.00 cm0.00 cm (Exact equality)
Trial 48.0 cm17.0 cm15.00 cm15.00 cm15.00 cm0.00 cm (Exact equality)
Trial 57.0 cm25.0 cm24.00 cm24.00 cm24.00 cm0.00 cm (Exact equality)

Tap a highlighted cell to write your own observation.

Result and conclusion

Tangents PT1 and PT2 from point P measure exactly 12.0 cm each. Triangle OPT1 is congruent to Triangle OPT2 by RHS criterion, proving PT1 = PT2.

PT1 = PT2 = √(OP² - r²) = √(13.0² - 5.0²) = √(169 - 25) = √144 = 12.0 cm (Tangents from external point are equal)
ObservationWhat it means
Congruence of Right TrianglesRight triangles ΔOPT1 and ΔOPT2 are congruent by RHS criterion (OT1 = OT2 = r, OP common hypotenuse, ∠OT1P = ∠OT2P = 90°).
Equal Tangent Lengths VerifiedCongruence directly implies PT1 = PT2 = 12.0 cm, confirming that lengths of tangents drawn from an external point are strictly equal.
Angle Subtension EqualityThe tangents subtend equal angles at the circle center (∠T1OP = ∠T2OP = 67.38°).
Line of Centers Bisects Tangent AngleLine segment OP bisects the angle between the two tangents (∠T1PO = ∠T2PO = 22.62°).
Distance OP from Center vs Resulting Tangent Length (PT)0611172205.0 cm (Point on circle)4.97.0 cm8.6610.0 cm1213.0 cm14.1415.0 cm19.3620.0 cmExternal Distance OP (cm)Tangent Length PT (cm) [for r = 5.0 cm]

Sample readings. Replace with your own results.

Experiment photos

More experiments