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Form 4 Chapter 8: Circles III

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A tangent to a circle

- a straight line which touches the circle at only one point

A B

AB = tangent to the circle

- perpendicular to the radius at the point of contact

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Constructing tangents to a circle

The diagram below shows a circle with centre O.Construct the tangent to the circle at the point P.

O

P

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O

P

Solution:

Draw a straight line joining point P and centre O.

Step 1:

Constructing tangents to a circle

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O

P

Solution:

Step 2:

Adjust your compasses so that its radius is slightly more than half of the length of OP.

Constructing tangents to a circle

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O

P

Solution:

Place your compasses at P. On the line OP, draw one point on both sides of P.

Step 3:

Constructing tangents to a circle

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O

P

Solution:

Place your compasses at one of the point on the line OP, draw an arc above and below the line OP.

Step 4:

Constructing tangents to a circle

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O

P

Solution:

With the same radius and another point on line OP as centre, draw another two arcs to intersect the ones drawn in step 4.

Step 5:

Constructing tangents to a circle

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O

P

Solution:

Join the two intersections with a straight line.

Step 6:

Tangent at P

Constructing tangents to a circle

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The diagram below shows a circle with centre O.Construct two tangents to the circle that pass through the point T.

Constructing tangents to a circle

OT

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OT

Solution:

Step 1:

Draw a straight line joining point T and centre O.

Constructing tangents to a circle

O

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OT

Solution:

Step 2:

Adjust your compasses so that its radius is slightly more than half of the length of OT.

Constructing tangents to a circle

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OT

Solution:

Step 3:Place your compasses at T. Draw a short arc above and below the line OT.

Constructing tangents to a circle

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OT

Solution:

With the same radius and your compasses placed at O, draw arcs to intersect the ones drawn in Step 3.

Step 4:

Constructing tangents to a circle

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OT

Solution:

Step 5:

Join the two intersections with a straight line.

Constructing tangents to a circle

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OT

Solution:

Step 6:

Label the midpoint as M. Using M as the centre and OM as the radius, draw two arcs that cut the circle at P and Q.

M

P

Q

Constructing tangents to a circle

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OT

Solution:

Step 7:

Join points P and Q to point T.

M

P

Q

Lines PT and QT are tangents to the

circle that pass through point T

Constructing tangents to a circle

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Properties related to two tangents to a circle

OT

P

Q

y°

y°

x°x°

PT = QT

PTO = OTQ

POT = QOT

∆POT and ∆QOT are congruent.

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