A few facts about circles answer most chord and tangent questions. A tangent touches the circle at one point and is perpendicular to the radius there, two tangents from the same outside point are equal in length, and the perpendicular from the centre to a chord cuts the chord in half.
Formulas
Tangent from an outside point, at distance d from the centre
tangent length=d2−r2
Chord at distance p from the centre
(2chord)2+p2=r2
Angles on the same arc
angle at the centre=2×angle at the circleangle in a semicircle=90∘
Worked examples
Example 1. A point P is 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P.
The tangent is perpendicular to the radius, so the radius, tangent and line to the centre form a right-angled triangle.
Tangent =132−52=169−25=144=12.
Answer: 12 cm
Example 2. A chord of length 16 cm is drawn in a circle of radius 10 cm. How far is it from the centre?
The perpendicular from the centre halves the chord: each half is 8 cm.
Distance =102−82=36=6.
Answer: 6 cm
Example 3. An arc subtends an angle of 110∘ at the centre. What angle does it subtend at a point on the rest of the circle?
The angle at the circle is half the angle at the centre.
110∘÷2=55∘.
Answer: 55∘
Example 4. PA and PB are tangents from P to a circle, with PA=7 cm and ∠APB=60∘. Find AB.
Tangents from the same point are equal, so PB=PA=7 cm and triangle PAB is isosceles.
Its base angles are (180∘−60∘)÷2=60∘ each, so it is equilateral: AB=7 cm.
Answer: 7 cm
Common mistakes
Adding instead of subtracting under the square root. For a tangent it is d2−r2: the line to the centre is the hypotenuse.
Using the full chord instead of half of it in (2chord)2+p2=r2.
Doubling instead of halving: the angle at the CENTRE is the larger one.
Practice questions with answers
1. What is the angle subtended at the center by a chord if the angle subtended at the circle is 40 degrees?
A80 degrees
B40 degrees
C20 degrees
D140 degrees
Show answer
Answer: A) 80 degrees
The central angle is twice the inscribed angle: 2×40=80 degrees.
2. What is a tangent to a circle?
AA line that intersects the circle at two points
BA line that touches the circle at exactly one point
CA line that passes through the center of the circle
DA line that is parallel to the circle
Show answer
Answer: B) A line that touches the circle at exactly one point
A tangent is defined as a line that touches a circle at exactly one point.
3. How many tangents can be drawn to a circle from a point lying ON the circle?
A0
B1
C2
DInfinitely many
Show answer
Answer: B) 1
At a point on the circle exactly one tangent can be drawn, perpendicular to the radius at that point.
4. What is the angle between a tangent to a circle and the radius drawn to the point of contact?
A180°
B60°
C90°
D45°
Show answer
Answer: C) 90°
The tangent at any point of a circle is perpendicular to the radius through the point of contact, so the angle is 90°.
5. The rim of a circular clock face carries the twelve hour markings, equally spaced around it. What angle does the chord joining the 12 marking to the 3 marking subtend at the centre of the clock?
A30 degrees
B90 degrees
C45 degrees
D120 degrees
Show answer
Answer: B) 90 degrees
The twelve markings share the full 360 degrees equally, so consecutive markings are 360 / 12 = 30 degrees apart at the centre. Going from 12 to 3 is three such steps, giving 3×30=90 degrees. The value 45 degrees is the angle this chord would subtend at a point on the major arc, which is half of 90.
6. A circular table of radius 60 cm is pushed across a room until its edge just meets a straight flat wall at a single point. Which statement correctly describes the situation at that point of contact?
AThe wall acts as a chord of the circular edge of the table
BThe wall acts as a tangent to the circular edge, and the radius drawn to the point of contact is perpendicular to the wall
CThe wall acts as a secant of the circular edge and cuts it at two points
DThe wall acts as a tangent, and the radius drawn to the point of contact makes an angle of 60 degrees with the wall
Show answer
Answer: B) The wall acts as a tangent to the circular edge, and the radius drawn to the point of contact is perpendicular to the wall
The wall meets the circular edge at exactly one point, which is precisely the definition of a tangent. By the tangent-radius theorem the radius drawn to the point of contact meets the tangent at 90 degrees, so the centre of the table is exactly 60 cm from the wall.
