2.4 Circles, Chords and Tangents

• Chords and tangents
• Lengths of tangents and intersecting chords
• Properties of chords and tangents
• Construction of tangents to a circle
• Direct and transverse common tangents to two circles
• Angles in the alternate segment
• Contact of circles
• Circumscribed, inscribed and escribed circles
• Centroid and orthocentre

• determine the length of intersecting chords and tangent to a circle
• construct a tangent to a circle at a given point
• construct direct and transverse common tangents to two circles
• determine the relationship of angles in the alternate segment
• construct an inscribed, circumscribed and escribed circles on a triangle
• determine the centroid and orthocentre of a triangle
• apply angles, chords and tangents to real life situations
• appreciate the use of circles, chords and tangents in real life situations

“How is the knowledge of tangents useful in day to day life?”

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1. The line from the centre to a chord
   40 min · 1 practice tasks · quiz
   The perpendicular from the centre to a chord bisects the chord, and Pythagoras then gives the missing length.
2. Two chords that cross
   40 min · 1 practice tasks · quiz
   When two chords cross inside a circle, the products of their pieces are equal.
3. A tangent meets the radius
   40 min · 1 practice tasks · quiz
   A tangent is perpendicular to the radius at the point of contact, so the tangent length comes from Pythagoras.
4. The two tangents from one point
   40 min · 1 practice tasks · quiz
   The two tangents from an external point are equal, and the line from the centre bisects the angles at that point and at the centre.
5. Drawing the tangents
   40 min · 1 practice tasks · quiz
   Construct the tangent at a point on the circle, and the two tangents from an external point, using the right angle in a semicircle.
6. A tangent and a secant
   40 min · 1 practice tasks · quiz
   If a tangent and a secant start at the same outside point, the square of the tangent equals the product of the whole secant and its external part.
7. Two secants from one point
   40 min · 1 practice tasks · quiz
   Two secants from the same outside point have equal products of external part and whole secant.
8. The alternate segment
   40 min · 1 practice tasks · quiz
   The angle between a tangent and a chord equals the angle subtended by that chord in the alternate segment.
9. A direct common tangent
   40 min · 1 practice tasks · quiz
   A direct common tangent does not cross between the circles, and its length is the square root of d squared minus (R - r) squared.
10. A transverse common tangent
    40 min · 1 practice tasks · quiz
    A transverse common tangent crosses between the circles, and its length uses the sum of the radii.
11. When two circles touch
    40 min · 1 practice tasks · quiz
    External contact adds the radii, internal contact uses their positive difference, and the centre distance fixes how many common tangents exist.
12. The circle through three vertices
    40 min · 1 practice tasks · quiz
    The perpendicular bisectors of the sides meet at the circumcentre, and in a right triangle that centre is the midpoint of the hypotenuse.
13. The incircle and an excircle
    40 min · 1 practice tasks · quiz
    Angle bisectors meet at the incentre, and two exterior bisectors give an excentre. The radii come from the area and the semiperimeter.
14. Centroid and orthocentre
    40 min · 1 practice tasks · quiz
    The medians meet at the centroid and divide each median in the ratio 2 to 1. The altitudes meet at the orthocentre.