A B C D Lesson 9.3

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

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

B. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse. A B C D h a b x y c x = h h y or h 2 = xy

C. 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 a 2 = ycorx = b or b 2 =xc a cb c A B C D h b xy a c

(CD) 2 = AD DB x 2 = 3 9 x = ± CD =

(AC) 2 = AD AB 10 2 = x(x + 21) x(x + 21) = x x – 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.

1.PK  JM 2.  PKJ is a rt.  3.  PKM is a rt.  4.RK  JP 5.RK is an altitude. 6.(PK) 2 = (PR)(PJ) 7.Similarly, (PK) 2 = (PO)(PM) 8.(PO)(PM) = (PR)(PJ) 1.Given 2.  segments form rt  s 3.Same as 2 4.Given 5.A segment drawn from a vertex of a Δ  to the opposite side of an altitude. 6.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. 7.Reasons Transitive Property