GEOMETRIA PRACTICE WITH CLIL The chord theorem Let us state and demonstrate the following theorem, already stated (page 132). THEOREM (of the chord) Within a circle, the relation between a chord and the sine of any of the peripheral angles referring to the same chord equals the diameter of the circumference. Demonstration K A B O F G Taken a chord AB, the peripheral angles on the circle, from the side of the major arc, are congruent, thus A BF A; the triangle BFA, having a side coincident with the diameter, has a right angle in B. Therefore, BG following the relations of the right triangles: A AB = AF sinBF AB AF = _______ A sinBF c.v.d. If, instead, we consider the peripheral angle referred to the chord AB, but from the side of the minor arc, since A = 180° BK A we have: BF AB AB AB AF = _______ = ______________ = _______ A sin(180° BK A) sinBK A sinBF which is again the thesis of the theorem. c.v.d. Exercises 1. Demonstrate that also the sine theorem results from this theorem. _ 2. Determine the length of the chord AB of a circle of radius r = 2, subtending an angle = _ . 4 3. Two consecutive chords AB and BC, of a circle of radius r = 15 cm, measure 22.03 cm and 27 cm respectively and the centre of the circumference is internal the angle ABC. Determine the perimeter and the area of the angle ABC. 170

PRACTICE WITH CLIL