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