altitudes–on- hypotenuse theorem

A B

C

D

Lesson 9.3

A BDC

AD C

B

D

A

C B

Three similar triangles: small, medium and large.

Altitude CD drawn to hyp. of △ABCThree 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

hab

x yc

x = hh y

or h2 = 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 a2 = yc or x = b or b2 =xc

a c b c

A B

C

D

hb

x y

a

c

(CD)2 = AD • DBx2 = 3 • 9x = ±x = ±CD =

(AC)2 = AD • AB102 = x(x + 21)x(x + 21) = 10 • 10x2 + 21x – 100 = 0(x + 25)(x – 4) = 0x + 25 = 0 OR x – 4 = 0x = -25 OR x = 4Since AD cannot be negative, AD

= 4.

1. PK JM2. PKJ is a rt. 3. PKM is a rt. 4. RK JP5. RK is an altitude.6. (PK)2 = (PR)(PJ)7. Similarly, (PK)2 =

(PO)(PM)8. (PO)(PM) = (PR)

(PJ)

1. Given2. segments form rt s3. Same as 24. Given5. 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 1-68. Transitive Property