Circles

• The Power Theorems illustrate the secants, tangents, segments, and sectors of a circle.
• The Power Theorems prove theorems on secants, tangents, and segments.
• The Power Theorems solve problems on circles.
• If the distance from the main road to Gate 2 is 70 m and the length of the pathway from Gate 2 to the Exit is 50 m, about how far from the main road is Gate 1?
• A satellite directly over the equator of Earth estimates that the angle formed by two tangents to the equator is 25°0.
• The measure of the part of the Earth that will be covered by the satellite as represented by arc AB can be determined by solving the problem.
• If the distance from B to D is 100,000 miles, the distance of AB can be determined by solving the problem.
• A secant segment is a segment that intersects a circle in two points, only one of these is an endpoint of the segment.
• A tangent segment is a segment that is part of a tangent line and one of its endpoints is the point of tangency.
• The lengths of the indicated segments can be found by completing the table.
• Circles are internally tangent if their centers are on the same side of the tangent line.
• Circles are externally tangent if their centers are on opposite sides of the tangent line.
• A common tangent is a line that is tangent to two circles.
• A common internal tangent is a common tangent that intersects the segment joining the centers of two circles.
• A common external tangent is a common tangent that does not intersect the segment joining the centers of two circles.
• The Tangent-Secant Segments Power Theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.
• In the given circle, if tangent and secant intersect at O, then (𝑪𝑶)𝟐 = 𝒀𝑶 ∙ 𝑵𝑶.
• The Segments of Secants Power Theorem:
• If two chords intersect in the interior of the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.
• If BS = 6, BC = x and CU = x + 1 then, what is the length of BU?
• If two secants intersect in the exterior of the circle, the product of the lengths of one secant segment and the length of its external part is equal to the product of the lengths of the other secant segment and its external part.
• Name the external tangent segments in each of the following figures.
• The Intersecting of Chords Power Theorem:
• If SE and EL intersect at L, then SL ● Il = ______ ● AL.
• Find the distance between the centers of the
• If AB = 4, BC = 6 and AE = 2 then DE = _______.
• Given below is Circle O and circle C. Is it possible for a line to be tangent to two or more circles at different points?
• If tangent VL and secant EL intersect at L, then (VL)2 = _______ ● ______
• If AC = 9, BC = 7 and DE = 3 then, AE = _______.
• If the chords SA and EI intersect at R, then SR ● _____ = ER ● ______.
• If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
• The belt fits tightly around two gears as shown below
• In the given figure, if secants intersect at the exterior point R, then AR • IR = NR • ER
• If BC = 8 and CU = 10, then, find the value of BS.