11.1.2 Arcs and Chords (continued)
The measure of a central angle is determined in this example. The two circles are given to be congruent and ≅ BC QR . Since congruent arcs have congruent central angles, m ∠ BAC = m ∠ QPR . Substitute the algebraic expressions into the equation. Solve for x . Substitute x = 7 into the expression for m ∠ BAC . The solution yields m ∠ BAC = 80 ° .
Example 4 Using Radii and Chords
If a radius or diameter of a circle is perpendicular to a chord, then it bisects the chord and its arc.
In a circle, the perpendicular bisector of a chord is a radius or diameter.
The measure of a chord is determined in this example. The length of the radius of the circle is given. It is given that PQ is perpendicular to RS . It is also given that PQ is divided into lengths of 8 and 2 units by the intersection of RS . The trick to solving this example is to draw radii, the lengths of which are known, to R and S . Because PQ and RS are perpendicular, PQ bisects RS , and RW and SW are equal. Use the Pythagorean Theorem to determine RW (or SW ). Substituting the known lengths gives RW = 6. Since RW and SW are equal, RS = 2(6) = 12.
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