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Altitudes–On- Hypotenuse Theorem

C. B. A. D. Altitudes–On- Hypotenuse Theorem. Lesson 9.3. A. B. D. Altitude CD drawn to hyp. of △ ABC. C. Three similar triangles are formed. A. C. B. D. D. A. C. B. C. Three similar triangles: small, medium and large. C. B. A. D. Theorem 68 :.

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Altitudes–On- Hypotenuse Theorem

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  1. C B A D Altitudes–On- Hypotenuse Theorem Lesson 9.3

  2. A B D Altitude CD drawn to hyp. of △ABC C Three similar triangles are formed. A C B D D A C B C Three similar triangles: small, medium and large.

  3. C B A D Theorem 68: If an altitude is drawn to the hypotenuse of a right triangle then, • The two triangles formed are similar to the given right triangle and to each other. Δ ADC ~ Δ ACB ~ Δ CDB

  4. C B A D • The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse. x = h or h2 = xy h y a b h y x c

  5. C B A D • Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg (ie…. the projection of that leg on the hypotenuse) y = a or a2 = ycorx = b or b2 =xc a c b c b a h x y c

  6. If AD = 3 and DB = 9, find CD. (CD)2 = AD • DB x2 = 3 • 9 x = ± x = ± CD =

  7. If DB = 21 and AC = 10, find AC. (AC)2 = AD • AB 102 = x(x + 21) x(x + 21) = 10 • 10 x2 + 21x – 100 = 0 (x + 25)(x – 4) = 0 x + 25 = 0 OR x – 4 = 0 x = -25 OR x = 4 Since AD cannot be negative, AD = 4.

  8. Given •  segments form rt s • Same as 2 • Given • A segment drawn from a vertex of a Δ  to the opposite side of an altitude. • If the altitude is drawn to the hypotenuse of a rt. Δ, then either leg of a given rt. Δ is the mean proportional between the hypotenuse adjacent to the leg. • Reasons 1-6 • Transitive Property • PK  JM • PKJ is a rt.  • PKM is a rt.  • RK  JP • RK is an altitude. • (PK)2 = (PR)(PJ) • Similarly, (PK)2 = (PO)(PM) • (PO)(PM) = (PR)(PJ)

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