The right angle lives at contact
If a tangent touches a circle at T and O is the centre, then OT is perpendicular to the tangent. The radius must end at the point of contact.
Learning goals
- State and represent the perpendicular-radius tangent theorem.
- Use the right angle at contact to find angles and lengths.
- Reproduce the textbook proof with a valid contradiction argument.
Concept lesson
Tangent and contact point
A tangent has exactly one common point T with the circle. Join centre O to this same point before applying the theorem.
Right-triangle relation
In a right triangle the side opposite 90 degrees is the hypotenuse and its square equals the sum of the squares of the other sides.
Theorem 10.1
The tangent at any point of a circle is perpendicular to the radius through the point of contact. Thus OT perpendicular PT and angle OTP equals 90 degrees.
Why perpendicularity must hold
Assume a different perpendicular OM meets the tangent at M. The perpendicular is the shortest distance, so OM is less than OT. But OT is a radius, making M lie inside the circle; the tangent would then meet the circle twice. This contradiction gives OM=OT and M=T.
Find tangent length from centre distance
For radius OT=5 cm and OP=13 cm, triangle OTP is right at T. Therefore PT=sqrt(OP squared minus OT squared)=sqrt(169-25)=12 cm.
Mark the usable 90 degree angle
In rotated tangent diagrams, first identify contact point T, draw OT, mark 90 degrees, and only then use angle sum or complementary angles.
Choose the hypotenuse correctly
Label OP as the hypotenuse because it is opposite the right angle at contact; select addition or subtraction in Pythagoras according to the unknown side.
Build the contradiction proof
Arrange the proof cards: assume another perpendicular, use shortest distance, place the foot inside the circle, obtain a second intersection, contradict tangency, conclude the foot is the contact point.
Radius and tangent in a composite figure
Extract the right triangle from a composite circle diagram, state the perpendicular theorem, then combine angle relations or Pythagoras without treating an unmarked segment as a radius.
Theorem proof and numerical answer protocol
For a proof, write the construction and contradiction explicitly. For a numerical problem, name the right angle, write the equation, substitute, calculate and attach units.
Perpendicular-radius mastery
Complete a theorem statement, a rotated angle item, a tangent-length calculation and the logical spine of Theorem 10.1 proof.