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Class 10 Circles: cheat sheet
CBSE Mathematics. Tangents to a circle: a tangent is at right angles to the radius, and the two tangents from an outside point are equal.
Key ideas
- A tangent meets a circle at exactly one point, the point of contact; a secant meets it at two points.
- From a point inside a circle there is no tangent, from a point on it one, and from a point outside it two.
- The tangent at any point is perpendicular to the radius through the point of contact.
- The two tangents from an outside point are equal in length.
- The line from the outside point to the centre bisects the angle between the two tangents.
- A quadrilateral drawn around a circle has AB + CD = AD + BC.
- Most numerical questions are Pythagoras in the right triangle made by the radius, the tangent and the line to the centre.
Remember
| Tangent length | PT = √(OP² − r²) |
| Angle between tangents | ∠APB + ∠AOB = 180° |
| Circumscribed quadrilateral | AB + CD = AD + BC |
Watch out
- Using OP² = PT² − r² (the hypotenuse is OP, the line to the centre).
- Thinking the tangent passes through the centre.
- Forgetting that both tangents from the same point are equal when splitting the sides of a triangle.
Worked example
From a point P, 50 cm from the centre O of a circle of radius 40 cm, tangents PA and PB are drawn. Find the length of AB.
Working: PA = 30; OP bisects AB at right angles, and (1/2) × OA × PA = (1/2) × OP × (AB/2), so AB = 2 × 40 × 30/50
Answer: 48 cm
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