10.5 Chords and Angle Relationships
Notes
- In the same circle, congruent chords have congruent arcs and are the same distance from the center.
- A diameter perpendicular to a chord bisects the chord and its arc.
- Intersecting chords: if chords cross inside, $a \cdot b = c \cdot d$ (products of the pieces). The angle formed is half the sum of the intercepted arcs.
- Angle outside the circle (two secants, two tangents, or one of each): half the difference of the intercepted arcs.
- Tangent and chord at the point of tangency: the angle is half the intercepted arc.
Practice
Two chords intersect inside a circle. One is split into pieces of 4 and 9; the other into pieces of 6 and $x$. Find $x$.
$4 \cdot 9 = 6x$, so $x = 6$.
Two chords intersect inside a circle, intercepting arcs of $70^\circ$ and $110^\circ$ (the arcs facing the angle and its vertical angle). Find the angle.
$\displaystyle \frac{70 + 110}{2} = 90$.
Two secants meet outside a circle. They intercept a far arc of $150^\circ$ and a near arc of $40^\circ$. Find the angle.
$\displaystyle \frac{150 - 40}{2} = 55$.
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