How to Construct the Tangent Line to a Circle
A line tangent to a circle touches the circle at exactly one point. The tangent line is perpendicular to the radius of the circle.
In maths problems, one can encounter either of two options: constructing the tangent from a point outside of the circle, or constructing the tangent to a circle at a point on the circle.
Constructing a Tangent Line from a Point Outside of the Circle
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Setup of the problem. Construct a line, tangent to the circle, passing through point P{\displaystyle P}.
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Connect the point P{\displaystyle P} with the centre of the circle. You must first find the centre of the circle if it has not been given to you.
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Bisect OP{\displaystyle OP}. The bisector intersects OP{\displaystyle OP} in A{\displaystyle A}.
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Construct a circle with radius AP{\displaystyle AP}, centred at A{\displaystyle A}. This circle intersects the original circle at points B{\displaystyle B} and C{\displaystyle C}.
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Connect P{\displaystyle P} with B{\displaystyle B} or C{\displaystyle C}. Both PB{\displaystyle PB} an PC{\displaystyle PC} are tangent to the circle.
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Erase any construction lines, if needed.
Constructing a Tangent Line to a Circle at a Point on the Circle
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Setup of the problem. Construct a line, tangent to the circle at P{\displaystyle P}.
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Draw the radius MP{\displaystyle MP}. You must first find the centre of the circle if it has not been given to you.
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Extend the radius past the circle.
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Construct the perpendicular to the radius through point P{\displaystyle P}. That perpendicular is the tangent to the circle at point P{\displaystyle P}.
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Erase any construction lines, if needed.
Things You'll Need
- Paper
- Compass
- Ruler
- Pencil and/or pen
- Eraser
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