Tangents, Radii and the Alternate Segment Theorem
Use the right angle between a tangent and the radius at the point of contact, the equal tangents from an outside point, and the alternate segment theorem.
What a learner can do afterwards
- Mark the right angle where a tangent meets the radius and use it
- Use the two equal tangents from an external point to find a length
- Apply the alternate segment theorem to find an angle in a triangle inside the circle
1 · Read
A tangent line meets a circle at one point. The radius ending at that contact point is perpendicular to the tangent: they meet at 90 degrees. The converse also holds. A line through a point on the circle is tangent there if it is perpendicular to the radius ending there. The rule concerns the contact radius, not every radius.
Tangent segments PA and PB from one outside point P to the same circle have equal lengths. Join centre O to A, B and P. Right triangles OAP and OBP share hypotenuse OP and have equal radius legs, so they are congruent. Thus PA = PB. To find a perimeter, add both tangents and the chord. If the two tangents total L, each is L / 2.
For tangent PT and contact radius OT, triangle OTP is right-angled at T. OP is the hypotenuse. Pythagoras gives OP^2 = OT^2 + PT^2. Add the squared legs to find OP. Subtract a squared leg from OP^2 to find the other leg. Take the positive square root for a length. For OT = 7 cm and PT = 24 cm, OP^2 = 49 + 576 = 625, so OP = 25 cm.
For chord AB and point C on the circle, choose tangent ray AX opposite C across line AB. The alternate segment theorem gives angle XAB = angle ACB. The other tangent ray gives the supplementary angle, totalling 180 degrees with angle XAB. To finish a triangle, subtract two known angles from 180. To split a tangent-radius right angle, subtract a known part from 90. Triangle angles 47 and 68 leave 65 degrees.
Radii OA and OB make triangle OAB isosceles. Its base angles are equal: each is (180 - angle AOB) / 2. At A, radius OA is perpendicular to the tangent. If chord AB lies inside that right angle, subtract angle OAB from 90 to find the tangent-chord angle. For a centre angle of 110 degrees, each base angle is 35 degrees, leaving a tangent-chord angle of 55 degrees.
Calculate a base angle first, then the remaining part of the tangent-radius right angle.
Tangents PA and PB form quadrilateral OAPB with centre O. Its angles at A and B are each 90 degrees. A quadrilateral's interior angles total 360 degrees, so its interior angles AOB and APB total 180 degrees. Subtract the outside-point angle from 180 to find this centre angle.
Mark the tangent right angle, match tangent pairs, copy alternate segment angles, then finish with sums.
Lesson, questions and flashcards written for this dataset.
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More to read
Check the contact radius
A tangent-radius argument needs a particular radius. If the line touches the circle at T, join T to the centre O. OT is perpendicular to the tangent. Drawing a different radius does not give the same right-angle guarantee.
The converse is useful when a question asks whether a line is tangent. Check two facts: the line passes through T on the circle, and the line is perpendicular to OT. Together they establish tangency at T. A right angle at a different point is not enough.
Choose the length relation
Check the endpoints before equating tangent lengths. The equal-tangent theorem concerns two segments from one outside point to one circle. Segments starting at different outside points need not match.
For a radius-tangent triangle, locate the contact point before using Pythagoras. That is the right-angle vertex. The side joining the centre to the outside point is the hypotenuse. Its square equals the sum of the squared radius and tangent length.
Finding the hypotenuse requires addition of squares. Finding either leg requires subtraction from the hypotenuse squared. Finish with the positive square root, then check that the hypotenuse is the longest side.
Two shapes made by radii
A chord and the radii to its endpoints form an isosceles triangle. Its two base angles are equal. Once the centre angle is known, subtract it from 180 degrees and divide the remainder equally between the base angles.
A tangent at an endpoint adds a right angle with the radius there. If the chord splits that right angle, subtract the triangle's base angle from 90 degrees to find the remaining tangent-chord angle.
Two tangents from P make quadrilateral OAPB when O is the centre and A and B are the contacts. Two corners are right angles. The remaining interior angles, at O and P, therefore total 180 degrees.
Choose the tangent ray
Naming a tangent line alone does not select an angle. At its contact point, the line has two opposite rays. A chord makes supplementary angles with those rays.
Let AB be the chord and C another point on the circle. Choose tangent ray AX on the opposite side of line AB from C. The alternate segment theorem gives angle XAB = angle ACB. If the question instead asks for the angle with the other tangent ray, subtract angle XAB from 180 degrees.
After recording the theorem's equality, identify the next shape. A triangle may require a 180-degree sum. A split tangent-radius right angle requires a 90-degree sum.
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