10. 8 The Power Theorems Lesson Objectives: After studying this section, you will be able to: Apply the “Power Theorems” related to circles (there are three of them!) The power theorems are applied when you need to find lengths of segments related to circles. All of the power theorems can be proven by introducing lines and setting up proportions between similar triangles. Theorem 95: (The CHORD-CHORD Power Theorem) - If two chords of a circle intersect inside a 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. V Example by Proof: Given ⊙O Conclusion: ΔLVE ~ ΔNSE EV • EN = EL• SE 1. 2. 3. 4. 5. ⊙O ∡V ∡S ∡L ∡N ΔLVE ~ Δ NSE 1. 2. 3. 4. 5. 6. EV • EN = EL • SE S E Given If two inscribed ∡s intercepts same arc, Same as 2 AA~ (2, 3) Ratios of corr. sides of ~Δs are O L N 6. Means-Extremes Products Theorem Notice that the two triangles formed above by the introduction of the dashed lines MUST be similar since they have two pair of angles inscribed on the same arcs. They also have a pair of congruent vertical angles! The two triangles are similar by AA~ (and by “No Choice,” AAA~) This means that the proportion you can write using the measures of two sides of similar triangles that result when two chords intersect in a circle can be expressed this way: Defining our variables in the table below: Similar Triangles Small Δ Large Δ Whether Short sides Long sides a b c d a c b d , they all result in the following product! a•d=b•c Summary: When two chords intersect in a circle, the products of the parts are equal! Hint: Try assigning values to segments a, b, c, and d! Let a = 3, b = 2, c = 6, and d = 4; a = 2, b = 3, c = 9, and d = 6… ! Theorem 96: (The Tangent-Secant Power Theorem) If a tangent segment and secant segment are drawn to a circle from an external point, then the square of the tangent segment equals the product of the entire secant segment times its external part. secant segment external part tangent segment Notice that introduction of the dashed segments again form similar triangles. Since the tangent is a side of both triangles, it becomes a geometric mean between the secant and its external segment. Therefore, the following proportion is possible: and the cross products of the proportion results in the following formula: tan2= (secant) (external part) Theorem 97: (The Secant-Secant Power Theorem) If two secant segments are drawn from an external point to a circle, then the product of the first secant segment and its external part equals the product of the second secant segment and its external part. secant 2 external 2 P secant 1 external 1 Again by introducing the dashed segments we end up with similar triangles where the proportion using corresponding sides is as follows: The cross products of this proportion result in the following formula: (SEC1 ) (Ext1) = (SEC2 ) (Ext2) Also, the term “power” indicates the product of factors. The factors are determined solely by an external point “P.” For a fixed point, “P,” any line through “P” determines two distances to the circle. The product of the distances is constant! Power Theorem Formulas: Type Chord-Chord Tangent-Secant Secant-Secant Components with Circle Formula two intersecting chords tangent segment & secant segment Two secant segments (part1)(part1) = (part2)(part2) (tan)2 = (part secant)(whole secant) (part1)(whole1) = (part2)(whole2)